text: ◦ ◦ ν event no. [1] δχ2 Energy (EeV) RA ( ) Dec ( ) − 1 0.41 108.2 45.6 1.7 − 2 0.002 80.9 283.7 28.6 − 7 0.85 83.8 26.8 24.8 − 11 0.10 100.1 150.1 10.3 − 15 0.6 82.3 287.7 64.9 − 16 0.006 127.1 192.8 21.2 − 0.52 84.7 199.7 34.9 − 21 0.55 111.8 352.6 20.8 − 21 0.6 83.8 26.8 24.8 − 22 0.85 80.9 283.7 28.6 − 24 0.77 80.9 283.7 28.6 − 25 0.095 80.2 283.7 28.6 0.12 118.3 287.7 1.5 0.61 82 299 19.4 0.62 80.2 271.1 19.0 0.67 81.4 308.8 16.1 33 0.34 82 287.7 1.5 0.95 82 299 19.4 34 0.22 81.4 308.8 16.1 0.34 82 299. 19.4 0.34 118.3 340.6 12 0.6 89 349.9 9.3 − 35 0.96 89.1 218.8 70.8 Table 3. IceCube cosmic neutrino events correlated with UHECRs detected by PAO above 80 EeV. ◦ conservative source search region of 3 around the directions of UHECRs which are correlated with cosmic neutrino events. We also limit our source search within a comoving volume with its radius set by the GZK eﬀect of p + γ interactions and corresponding energy losses by UHECR protons UHECR CMB [5, 6]. A crude estimate of the mean-free-path for this interaction can be obtained from the number density of CMB photons with 2.73 K temperature in the local universe, which is × 1.32 104((cid:15)/meV)3 − (cid:15)n((cid:15)) = − cm 3. (4.2) exp [4.25((cid:15)/meV)] 1 − Thus the number density is 244 cm 3 at the peak photon energy (cid:15) = 2.82k (2.73 K) = × − − B 0.66 meV, where k = 8.62 10 5 eV K 1 is the Boltzmann constant. A parametrization B of the UHECR proton’s mean-free-path, using delta function approximation of the pγ cross (cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21) section, is given by − ≈ E 3 E − λ 245.76 p exp 0.42 p 1 Mpc, (4.3) p 100 EeV 100 EeV which reproduces results from numerical calculations with accurate treatment [21] within ∼ ∼ 10% in the 60–200 EeV range. For reference, λ = 539, 247, 138 and 26 Mpc at p – 12 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: ν

text: event

text: no.

text: δχ2

text: Energy

text: (EeV)

text: RA

text: Dec

text: Table

text: IceCube

text: cosmic

text: neutrino

text: events

text: correlated

text: with

text: UHECRs

text: detected

text: by

text: PAO

text: above

text: EeV.

text: conservative

text: source

text: search

text: region

text: of

text: around

text: the

text: directions

text: of

text: UHECRs

text: which

text: are

text: correlated

text: with

text: cosmic

text: neutrino

text: events.

text: We

text: also

text: limit

text: our

text: source

text: search

text: within

text: a

text: comoving

text: volume

text: with

text: its

text: radius

text: set

text: by

text: the

text: GZK

text: eﬀect

text: of

text: p

text: γ

text: interactions

text: and

text: corresponding

text: energy

text: losses

text: by

text: UHECR

text: protons

text: UHECR

text: CMB

text: A

text: crude

text: estimate

text: of

text: the

text: mean-free-path

text: for

text: this

text: interaction

text: can

text: be

text: obtained

text: from

text: the

text: number

text: density

text: of

text: CMB

text: photons

text: with

text: K

text: temperature

text: in

text: the

text: local

text: universe,

text: which

text: is

text: cm

text: exp

text: Thus

text: the

text: number

text: density

text: is

text: cm

text: at

text: the

text: peak

text: photon

text: energy

text: K)

text: B

text: meV,

text: where

text: k

text: eV

text: K

text: is

text: the

text: Boltzmann

text: constant.

text: A

text: parametrization

text: B

text: of

text: the

text: UHECR

text: proton’s

text: mean-free-path,

text: using

text: delta

text: function

text: approximation

text: of

text: the

text: pγ

text: cross

text: section,

text: is

text: given

text: by

text: E

text: E

text: λ

text: p

text: exp

text: p

text: Mpc,

text: p

text: EeV

text: EeV

text: which

text: reproduces

text: results

text: from

text: numerical

text: calculations

text: with

text: accurate

text: treatment

text: within

text: in

text: the

text: EeV

text: range.

text: For

text: reference,

text: λ

text: and

text: Mpc

text: at

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − · γ = cos 1(xˆ xˆ ), (3.1) neutrino UHECR is an invariant measure of the angular correlation between the neutrino and UHECR arrival directions [7, 8]. Following ref. [7] we use a statistic made from invariant γ for each neutrino direction xˆ and UHECR direction xˆ pair as i j δχ2 = min (γ2 /δγ2), (3.2) i j ij i which is minimized for all j. Here δγ is the 1-σ angular resolution of the neutrino events. i We use the exact resolutions reported by the IceCube collaboration for each event [1]. ≤ A value δχ2 1 is considered a “good match” between the i-th neutrino and an UHECR i arrival directions. We exploit distributions of all δχ2 statistic to study angular correlation i between IceCube neutrino events and UHECR data. The distribution with observed data ≤ giving a number of “hits” or N with δχ2 1 therefore forms a basis to claim correlation. hits Note that in case more than one UHECR directions are within the error circle of a neutrino event, the δχ2 value for UHECR closest to the neutrino direction is chosen in this method. We estimate the signiﬁcance of any correlation in data by comparing N with corre- hits sponding number from null distributions. We construct two null distributions, in one case we randomize only the RA of UHECRs, keeping their Dec the same as in data; and in the second case we also randomize Dec according to the zenith-angle depended sky exposure of the UHECR experiments [36], aﬀecting the declination distributions of UHECR data. We call these two null distributions as the semi-isotropic null and exposure-corrected null, respectively. The semi-isotropic null is a quick-way to check signiﬁcance while the exposure-corrected null is accurate when information on particular experiments are available. In both cases we perform 100,000 realizations of drawing random numbers to assign new RA and Dec values for each event to construct δχ2 distributions in the same way as done with real data. We ﬁnd that the two null distributions are in good agreement with each other in most cases. We calculate statistical signiﬁcance of correlation in real data or p-value (chance proba- bility) using frequentists’ approach. We count the number of times we get a random data set ≤ that gives equal or more hits than the N in real data within δχ2 1 bin. Dividing this hits number with the total number of random data sets generated (100,000) gives us the p-value. We cross-check this p-value by calculating the Poisson probability of obtaining N within ≤ hits δχ2 1 bin given the corresponding average hits expected from the null distribution. These two chance probabilities are in good agreement. 4 Results 4.1 Correlations between neutrinos and UHECRs We apply our statistical method separately to the PAO and TA data, to a combination of the both and to all available UHECR data above 100 EeV from all experiments (see Table ≤ ≤ 1). The results are shown in the histograms of Figure 3. The counts in the 0 δχ2 1 bins for the blue, ﬁlled histograms correspond to the number of correlated neutrino events with UHECRs. Counts in other bins are due to distant pairs of the neutrino events and UHECRs and are uninteresting for us. The counts for the red (green), open histogram in the same bins correspond to the expected number of correlated neutrino events from the semi-isotropic – 6 –

label: equation

text: γ

text: cos

text: xˆ

text: neutrino

text: UHECR

text: is

label: paragraph

text: an

text: invariant

text: measure

text: of

text: the

text: angular

text: correlation

text: between

text: the

text: neutrino

text: and

text: UHECR

text: arrival

text: directions

text: Following

text: ref.

text: we

text: use

text: a

text: statistic

text: made

text: from

text: invariant

text: γ

text: for

text: each

text: neutrino

text: direction

text: xˆ

text: and

text: UHECR

text: direction

text: xˆ

text: pair

text: as

text: i

text: j

text: δχ2

text: min

text: i

text: j

text: ij

text: i

text: which

text: is

text: minimized

text: for

text: all

text: j.

text: Here

text: δγ

text: is

text: the

text: angular

text: resolution

text: of

text: the

text: neutrino

text: events.

text: i

text: We

text: use

text: the

text: exact

text: resolutions

text: reported

text: by

text: the

text: IceCube

text: collaboration

text: for

text: each

text: event

text: A

text: value

text: δχ2

text: is

text: considered

text: a

text: “good

text: match”

text: between

text: the

text: i-th

text: neutrino

text: and

text: an

text: UHECR

text: i

text: arrival

text: directions.

text: We

text: exploit

text: distributions

text: of

text: all

text: δχ2

text: statistic

text: to

text: study

text: angular

text: correlation

text: i

text: between

text: IceCube

text: neutrino

text: events

text: and

text: UHECR

text: data.

text: The

text: distribution

text: with

text: observed

text: data

text: giving

text: a

text: number

text: of

text: “hits”

text: or

text: N

text: with

text: δχ2

text: therefore

text: forms

text: a

text: basis

text: to

text: claim

text: correlation.

text: hits

text: Note

text: that

text: in

text: case

text: more

text: than

text: one

text: UHECR

text: directions

text: are

text: within

text: the

text: error

text: circle

text: of

text: a

text: neutrino

text: event,

text: the

text: δχ2

text: value

text: for

text: UHECR

text: closest

text: to

text: the

text: neutrino

text: direction

text: is

text: chosen

text: in

text: this

text: method.

text: We

text: estimate

text: the

text: signiﬁcance

text: of

text: any

text: correlation

text: in

text: data

text: by

text: comparing

text: N

text: with

text: corre-

text: hits

text: sponding

text: number

text: from

text: distributions.

text: We

text: construct

text: two

text: distributions,

text: in

text: one

text: case

text: we

text: randomize

text: only

text: the

text: RA

text: of

text: UHECRs,

text: keeping

text: their

text: Dec

text: the

text: same

text: as

text: in

text: data;

text: and

text: in

text: the

text: second

text: case

text: we

text: also

text: randomize

text: Dec

text: according

text: to

text: the

text: zenith-angle

text: depended

text: sky

text: exposure

text: of

text: the

text: UHECR

text: experiments

text: aﬀecting

text: the

text: declination

text: distributions

text: of

text: UHECR

text: data.

text: We

text: call

text: these

text: two

text: distributions

text: as

text: the

text: semi-isotropic

text: and

text: exposure-corrected

text: null,

text: respectively.

text: The

text: semi-isotropic

text: is

text: a

text: quick-way

text: to

text: check

text: signiﬁcance

text: while

text: the

text: exposure-corrected

text: is

text: accurate

text: when

text: information

text: on

text: particular

text: experiments

text: are

text: available.

text: In

text: both

text: cases

text: we

text: perform

text: realizations

text: of

text: drawing

text: random

text: numbers

text: to

text: assign

text: new

text: RA

text: and

text: Dec

text: values

text: for

text: each

text: event

text: to

text: construct

text: δχ2

text: distributions

text: in

text: the

text: same

text: way

text: as

text: done

text: with

text: real

text: data.

text: We

text: ﬁnd

text: that

text: the

text: two

text: distributions

text: are

text: in

text: good

text: agreement

text: with

text: each

text: other

text: in

text: most

text: cases.

text: We

text: calculate

text: statistical

text: signiﬁcance

text: of

text: correlation

text: in

text: real

text: data

text: or

text: p-value

text: (chance

text: proba-

text: bility)

text: using

text: frequentists’

text: approach.

text: We

text: count

text: the

text: number

text: of

text: times

text: we

text: get

text: a

text: random

text: data

text: set

text: that

text: gives

text: equal

text: or

text: more

text: hits

text: than

text: the

text: N

text: in

text: real

text: data

text: within

text: δχ2

text: bin.

text: Dividing

text: this

text: hits

text: number

text: with

text: the

text: total

text: number

text: of

text: random

text: data

text: sets

text: generated

text: gives

text: us

text: the

text: p-value.

text: We

text: cross-check

text: this

text: p-value

text: by

text: calculating

text: the

text: Poisson

text: probability

text: of

text: obtaining

text: N

text: within

text: hits

text: δχ2

text: bin

text: given

text: the

text: corresponding

text: average

text: hits

text: expected

text: from

text: the

text: distribution.

text: These

text: two

text: chance

text: probabilities

text: are

text: in

text: good

text: agreement.

text: Results

text: Correlations

text: between

text: neutrinos

text: and

text: UHECRs

text: We

text: apply

text: our

text: statistical

text: method

text: separately

text: to

text: the

text: PAO

text: and

text: TA

text: data,

text: to

text: a

text: combination

text: of

text: the

text: both

text: and

text: to

text: all

text: available

text: UHECR

text: data

text: above

text: EeV

text: from

text: all

text: experiments

text: (see

text: Table

text: The

text: results

text: are

text: shown

text: in

text: the

text: histograms

text: of

text: Figure

text: The

text: counts

text: in

text: the

text: δχ2

text: bins

text: for

text: the

text: blue,

text: ﬁlled

text: histograms

text: correspond

text: to

text: the

text: number

text: of

text: correlated

text: neutrino

text: events

text: with

text: UHECRs.

text: Counts

text: in

text: other

text: bins

text: are

text: due

text: to

text: distant

text: pairs

text: of

text: the

text: neutrino

text: events

text: and

text: UHECRs

text: and

text: are

text: uninteresting

text: for

text: us.

text: The

text: counts

text: for

text: the

text: red

text: (green),

text: open

text: histogram

text: in

text: the

text: same

text: bins

text: correspond

text: to

text: the

text: expected

text: number

text: of

text: correlated

text: neutrino

text: events

text: from

text: the

text: semi-isotropic


text: 12 TRUNCATIONS OF UNITARY ENSEMBLES × U Proposition 3.3. The 2 (cid:88)2 spectral mea(cid:88)sure of Sp(n) is n n W δ + WRδ − , (3.14) j eiθj j e iθj (cid:20) j=1 (cid:21) j=1 (cid:20) (cid:21) where √ √ − W = √ µj − µjνj eiψj , WR = −√ νj − µjνj eiψj . j µjνj e iψj νj j µjνj e iψj µj Here ≤ ≤ 0 < θ < π (1 j n), j (cid:88)≤ (cid:88) ≤ ≤ 0 ψ < 2π (1 j n), j n n ≤ ≤ µ + ν = 1, µ > 0, ν > 0 (1 j n), j j j j j=1 j=1 (cid:89) and the joint distribution of eiθ1, . . . , eiθn, eiψ1, . . . , eiψn, µ1, . . . , µn, ν1, . . . , νn−1 is | | n 2n ∆(2 cos θ , . . . , 2 cos θ ) 2 sin2 θ dθ . . . dθ n!πn 1 n j 1 n j=1 × × − dψ1 . . . dψn (2n 1)! dµ . . . dµ dν . . . dν − . 2π 2π 1 n 1 n 1 Proof. The distribution of the eigenvalues is a classical result. As for the matrix ∈ U eigenweights, ﬁrst write a matrix U Sp(n) as U = BDBR, where C(D) = − − diag(eiθ1, e iθ1, . . . , eiθn, e iθn). Repeating the arguments from the proof of Propo- U sition 3.1, we may think of B as being(cid:20)Haar dist(cid:21)ributed in Sp(n). Let the ﬁrst ∈ H quaternionic column of BR be [q , . . . , q ]T , q R, with 1 n j − α β¯ C(q ) = j j . j β α¯ j j (cid:88) (cid:88) Now Um = BDmBR and a simple calculation imply (cid:104) (cid:105) n | | n | | − e1, C(U)me1 = (cid:88) αj 2eimθj +(cid:88) βj 2e imθj, j=1 j=1 (cid:104) (cid:105) n | | n | | − e2, C(U)me2 = (cid:88)βj 2eimθj + (cid:88)αj 2e imθj, j=1 j=1 (cid:104) (cid:105) − n n − e1, C(U)me2 = (cid:88) α¯jβ¯jeimθj + (cid:88) α¯jβ¯je imθj, j=1 j=1 (cid:104) (cid:105) − n n − e , C(U)me = α β eimθj + α β e imθj. 2 1 j j j j j(cid:20)=1 j(cid:21)=1 Therefore the matrix-valued measure (2.7) has the form (3.14) with | | − α 2 α¯ β¯ W = − j | j| j . j α β β 2 j j j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRUNCATIONS

text: OF

text: UNITARY

text: ENSEMBLES

text: U

text: Proposition

text: The

text: spectral

text: of

text: Sp(n)

text: is

text: n

text: n

text: W

text: δ

text: WRδ

text: j

text: eiθj

text: j

text: e

text: iθj

text: where

text: W

text: µj

text: µjνj

text: eiψj

text: WR

text: νj

text: µjνj

text: eiψj

text: j

text: µjνj

text: e

text: iψj

text: νj

text: j

text: µjνj

text: e

text: iψj

text: µj

text: Here

text: θ

text: π

text: j

text: n),

text: j

text: ψ

text: 2π

text: j

text: n),

text: j

text: n

text: n

text: µ

text: ν

text: µ

text: ν

text: j

text: n),

text: j

text: j

text: j

text: j

text: and

text: the

text: joint

text: distribution

text: of

text: eiθn,

text: eiψn,

text: µn,

text: νn−1

text: is

text: n

text: cos

text: θ

text: cos

text: θ

text: θ

text: dθ

text: dθ

text: n!πn

text: n

text: j

text: n

text: dψn

text: dµ

text: dµ

text: dν

text: dν

text: 2π

text: 2π

text: n

text: n

text: Proof.

text: The

text: distribution

text: of

text: the

text: eigenvalues

text: is

text: a

text: classical

text: result.

text: As

text: for

text: the

text: matrix

text: U

text: eigenweights,

text: ﬁrst

text: write

text: a

text: matrix

text: U

text: Sp(n)

text: as

text: U

text: BDBR,

text: where

text: C(D)

text: e

text: eiθn,

text: e

text: iθn).

text: Repeating

text: the

text: arguments

text: from

text: the

text: proof

text: of

text: Propo-

text: U

text: sition

text: we

text: may

text: think

text: of

text: B

text: as

text: in

text: Sp(n).

text: Let

text: the

text: ﬁrst

text: H

text: quaternionic

text: column

text: of

text: BR

text: be

text: [q

text: q

text: ]T

text: q

text: R,

text: with

text: n

text: j

text: α

text: β¯

text: C(q

text: j

text: j

text: j

text: β

text: α¯

text: j

text: j

text: Now

text: Um

text: BDmBR

text: and

text: a

text: simple

text: calculation

text: imply

text: n

text: n

text: αj

text: βj

text: imθj,

text: n

text: n

text: imθj,

text: n

text: n

text: α¯jβ¯jeimθj

text: α¯jβ¯je

text: imθj,

text: n

text: n

text: e

text: C(U)me

text: α

text: β

text: eimθj

text: α

text: β

text: e

text: imθj.

text: j

text: j

text: j

text: j

text: Therefore

text: the

text: matrix-valued

text: measure

text: has

text: the

text: form

text: with

text: α

text: α¯

text: β¯

text: W

text: j

text: j|

text: j

text: j

text: α

text: β

text: β

text: j

text: j

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: TRUNCATIONS OF UNITARY ENSEMBLES 19 With this lemma and the results of Section in hand, we immediately obtain matrix models for the truncations of unitary ensembles. O SO O\ Indeed, choose a matrix U from one of the “scalar” ensembles from Section (CUE, COE, , , SO ), and remove any row and the corresponding column. By the invariance under (cid:0) (cid:1) the group action, it does not matter which row and column we delete. It will be convenient to delete the ﬁrst ones. Then the truncated matrix V is † (cid:104) (cid:105)⊥ V = PUP , where P = proj e . 1 C † † Recall now (2.5) that U = W W with We = W e = e which implies 1 1 1 C † † C † † V = PW W P = WP P W . C Therefore V is unitarily equivalent to the CMV matrix form of U but with the ﬁrst row and column removed. Thus the results of the next proposition follow from the CMV models in Section , the previous lemma, and the rotational (or real even) symmetry of the Verblunsky coeﬃcients. U For quaternionic ensembles CSE and Sp one needs to modify all of the argu- ments in the obvious way to accommodate quaternionic rows and columns. Proposition 5.2. (a) The eigenvalue distribution of a COE(n + 1) or CUE(n + 1) matrix with one row and column removed coincides with the eigenvalue distribution of C (α , . . . , α − ) with independent Verblunsky coeﬃcients 0 n 1 ∼ ≤ ≤ − α Θ(β(k + 1) + 1), 0 k n 1, (5.1) k where β = 1 for COE and β = 2 for CUE. The eigenvalue distribution of a CSE(n + 1) matrix with one (quater- nionic) row and column removed coincides with the eigenvalue distribution C of (α I , . . . , α − I ) where α are independent and distributed according 0 2 n 1 2 k to (5.1) with β = 4. O SO O \SO (b) The eigenvalue distribution of a (n+1), or (n+1), or (n+1) (n+ 1) matrix with one row(cid:0)and colum(cid:1)n removed coincides with the distribution C of (α , . . . , α − ) with independent Verblunsky coeﬃcients 0 n 1 ∼ ≤ ≤ − α B k+1, k+1 , 0 k n 1. k 2 2 U (c) The eigenvalue distribution of a Sp(n+1) matrix with one (quaternionic) C row and column removed coincides with the distribution of (α , . . . , α − ) × 0 n 1 with independent 2 2 matrix Verblunsky coeﬃcients ∼ ≤ ≤ − α Υ (4k + 7) , 0 k n 1. k Remarks. 1. From the aesthetic considerations, one might prefer to look at the − − − − above distributions as B(1(k+2) 1, 1(k+2) 1), Θ(2(k + 2) 1), Υ(4(k + 2) 1) for 2 O2 U U the truncations of the compact groups , , and Sp, respectively. Note B has an extra factor of 1 due to the historical deﬁnition of the Beta function. 2 2. These CMV matrices are no longer unitary as the last coeﬃcient is no longer unimodular (with probability 1), see the remark after Deﬁnition 2.3. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRUNCATIONS

text: OF

text: UNITARY

text: ENSEMBLES

text: With

text: this

text: lemma

text: and

text: the

text: results

text: of

text: Section

text: in

text: hand,

text: we

text: immediately

text: obtain

text: matrix

text: models

text: for

text: the

text: truncations

text: of

text: unitary

text: ensembles.

text: O

text: SO

text: O\

text: Indeed,

text: choose

text: a

text: matrix

text: U

text: from

text: one

text: of

text: the

text: “scalar”

text: ensembles

text: from

text: Section

text: (CUE,

text: COE,

text: SO

text: and

text: remove

text: any

text: row

text: and

text: the

text: corresponding

text: column.

text: By

text: the

text: invariance

text: under

text: the

text: group

text: action,

text: it

text: does

text: not

text: matter

text: which

text: row

text: and

text: column

text: we

text: delete.

text: It

text: will

text: be

text: convenient

text: to

text: delete

text: the

text: ﬁrst

text: ones.

text: Then

text: the

text: truncated

text: matrix

text: V

text: is

text: V

text: PUP

text: where

text: P

text: proj

text: e

text: C

text: Recall

text: now

text: that

text: U

text: W

text: W

text: with

text: We

text: W

text: e

text: e

text: which

text: implies

text: C

text: C

text: V

text: PW

text: W

text: P

text: WP

text: P

text: W

text: C

text: Therefore

text: V

text: is

text: unitarily

text: equivalent

text: to

text: the

text: CMV

text: matrix

text: form

text: of

text: U

text: but

text: with

text: the

text: ﬁrst

text: row

text: and

text: column

text: removed.

text: Thus

text: the

text: results

text: of

text: the

text: next

text: proposition

text: follow

text: from

text: the

text: CMV

text: models

text: in

text: Section

text: the

text: previous

text: lemma,

text: and

text: the

text: rotational

text: (or

text: real

text: even)

text: symmetry

text: of

text: the

text: Verblunsky

text: coeﬃcients.

text: U

text: For

text: quaternionic

text: ensembles

text: CSE

text: and

text: Sp

text: one

text: needs

text: to

text: modify

text: all

text: of

text: the

text: argu-

text: ments

text: in

text: the

text: obvious

text: way

text: to

text: accommodate

text: quaternionic

text: rows

text: and

text: columns.

text: Proposition

text: (a)

text: The

text: eigenvalue

text: distribution

text: of

text: a

text: COE(n

text: or

text: CUE(n

text: matrix

text: with

text: one

text: row

text: and

text: column

text: removed

text: coincides

text: with

text: the

text: eigenvalue

text: distribution

text: of

text: C

text: (α

text: α

text: with

text: independent

text: Verblunsky

text: coeﬃcients

text: n

text: α

text: Θ(β(k

text: k

text: n

text: k

text: where

text: β

text: for

text: COE

text: and

text: β

text: for

text: CUE.

text: The

text: eigenvalue

text: distribution

text: of

text: a

text: CSE(n

text: matrix

text: with

text: one

text: (quater-

text: nionic)

text: row

text: and

text: column

text: removed

text: coincides

text: with

text: the

text: eigenvalue

text: distribution

text: C

text: of

text: (α

text: I

text: α

text: I

text: where

text: α

text: are

text: independent

text: and

text: distributed

text: according

text: n

text: k

text: to

text: with

text: β

text: O

text: SO

text: O

text: \SO

text: (b)

text: The

text: eigenvalue

text: distribution

text: of

text: a

text: or

text: or

text: (n+

text: matrix

text: with

text: one

text: removed

text: coincides

text: with

text: the

text: distribution

text: C

text: of

text: (α

text: α

text: with

text: independent

text: Verblunsky

text: coeﬃcients

text: n

text: α

text: B

text: k

text: n

text: k

text: U

text: (c)

text: The

text: eigenvalue

text: distribution

text: of

text: a

text: matrix

text: with

text: one

text: (quaternionic)

text: C

text: row

text: and

text: column

text: removed

text: coincides

text: with

text: the

text: distribution

text: of

text: (α

text: α

text: n

text: with

text: independent

text: matrix

text: Verblunsky

text: coeﬃcients

text: α

text: Υ

text: k

text: n

text: k

text: Remarks.

text: From

text: the

text: aesthetic

text: considerations,

text: one

text: might

text: prefer

text: to

text: look

text: at

text: the

text: above

text: distributions

text: as

text: for

text: U

text: U

text: the

text: truncations

text: of

text: the

text: compact

text: groups

text: and

text: Sp,

text: respectively.

text: Note

text: B

text: has

text: an

text: extra

text: factor

text: of

text: due

text: to

text: the

text: historical

text: deﬁnition

text: of

text: the

text: Beta

text: function.

text: These

text: CMV

text: matrices

text: are

text: no

text: longer

text: unitary

text: as

text: the

text: last

text: coeﬃcient

text: is

text: no

text: longer

text: unimodular

text: (with

text: probability

text: see

text: the

text: remark

text: after

text: Deﬁnition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: TRUNCATIONS OF UNITARY ENSEMBLES 5 Deﬁnition 2.1. The un(cid:8)itary, orthogonal, special ortho(cid:9)gonal, and compact symplec- tic groups are deﬁned, r(cid:8)espectively, by (cid:9) U ∈ C × † † (n) = (cid:8)Q n n : Q Q = QQ = In , (cid:9) O ∈ R × (n) = (cid:8)Q n n : QT Q = QQT = In , (cid:9) SO ∈ R × (n) = Q n n : QT Q = QQT = I , det Q = 1 , n U ∈ H × † † Sp(n) = Q nR n : C(Q) C(Q) = C(Q)C(Q) = I . 2n × † Here I is the n n identity matrix; and T stands for the Hermitian conjugation n and transposition, respectively. By the general theory, each of these groups possesses the Haar measure, which is the unique (normalized by 1) measure that is invariant under the group action. U O SO U We will use the same notation (n), (n), (n), Sp(n) for the group itself and for the random ensemble. Next we describe the joint law of the columns of matrices from these groups. This is well known, as is the method of proof we employ. We give details only in U the Sp case where the argument contains an additional subtlety. O Proposition 2.2. (a) The ﬁrst column v of a matrix from (n) is distributed { ∈ R || || 1 } ≤ ≤ uniformly on the sphere x n : x = 1 . For 2 k n, the k-th column v is distributed uniformly on the subset of this unit sphere that is orthogonal to k v , . . . , v − . 1 k 1 U (b) The ﬁrst column v of a matrix from (n) is distributed uniformly on the { ∈ C || || 1 } ≤ ≤ sphere z n : z = 1 . For 2 k n, the k-th column v is distributed k uniformly on the subset of this unit sphere that is orthogonal to v , . . . , v − . ∈ U 1 k 1 (c) Let Q Sp(n). The ﬁrst column v of C(Q) is distributed uniformly on the { ∈ C || || } 1 sphere z 2n : z = 1 . The second column v is Zv¯ , where Z is (A.3). For ≤ ≤ − 2 1 2 k n, the (2k 1)-th column v − is distributed uniformly on the subset of 2k 1 this unit sphere that is orthogonal to v , . . . , v − , and v = Zv¯ − . 1 2k 2 2k 2k 1 U Proof. As noted above, we give details only for the case of Sp(n). Form V with columns v , . . . , v described by the procedure in the Proposition, 1 2n ∈ U and let us show that V is indeed C(Q) for a Q Sp(n). By the construction, C ⊥ v , . . . , v form an orthonormal basis of 2n since v Zv¯ holds for any vector 1 2n v. Thus V is unitary. Moreover, V consists of n2 blocks each of which has the − real quaternionic form, see (A.2). Therefore C 1(V ) belongs to the group (not U ensemble yet) Sp(n). Let us show that for any non-random W chosen from the U group C( Sp(n)), the distribution of V and WV are identical. This will prove that − C 1(V ) was generated according to the Haar measure by the uniqueness of the U measure invariant with respect to the group action on Sp(n). Denote the columns of WV by u , . . . , u . First note that u = Wv is a 1 2n { ∈ C || || 1 } 1 vector uniformly distributed on the unit sphere z 2n : z = 1 since v is. 1 The second column u = Wv = WZv¯ = ZWv = Zu¯ (we used (A.4)), which 2 2 1 1 1 agrees with our construction. Now inductively, consider u − , which is orthogonal × 2k 1 { } to u , . . . , u − . Consider any 2n 2n unitary U that maps span u , . . . , u − 1 2k 2 − { } 1 2k 2 to itself. Then W 1UW maps span v , . . . , v − to itself. By the construction − 1 2k 2 this means that v − and W 1UWv − are distributed identically. Thus u − = 2k 1 2k 1 2k 1 ##LTLine##

text: TRUNCATIONS

text: OF

text: UNITARY

text: ENSEMBLES

text: Deﬁnition

text: The

text: orthogonal,

text: special

text: and

text: compact

text: symplec-

text: tic

text: groups

text: are

text: deﬁned,

text: by

text: U

text: C

text: (n)

text: n

text: n

text: Q

text: Q

text: QQ

text: In

text: O

text: R

text: (n)

text: n

text: n

text: QT

text: Q

text: QQT

text: In

text: SO

text: R

text: (n)

text: Q

text: n

text: n

text: QT

text: Q

text: QQT

text: I

text: det

text: Q

text: n

text: U

text: H

text: Sp(n)

text: Q

text: nR

text: n

text: C(Q)

text: C(Q)

text: C(Q)C(Q)

text: I

text: Here

text: I

text: is

text: the

text: n

text: n

text: identity

text: matrix;

text: and

text: T

text: stands

text: for

text: the

text: Hermitian

text: conjugation

text: n

text: and

text: transposition,

text: respectively.

text: By

text: the

text: general

text: theory,

text: each

text: of

text: these

text: groups

text: possesses

text: the

text: Haar

text: measure,

text: which

text: is

text: the

text: unique

text: (normalized

text: by

text: measure

text: that

text: is

text: invariant

text: under

text: the

text: group

text: action.

text: U

text: O

text: SO

text: U

text: We

text: will

text: use

text: the

text: same

text: notation

text: (n),

text: (n),

text: (n),

text: Sp(n)

text: for

text: the

text: group

text: itself

text: and

text: for

text: the

text: random

text: ensemble.

text: Next

text: we

text: describe

text: the

text: joint

text: law

text: of

text: the

text: columns

text: of

text: matrices

text: from

text: these

text: groups.

text: This

text: is

text: well

text: known,

text: as

text: is

text: the

text: method

text: of

text: proof

text: we

text: employ.

text: We

text: give

text: details

text: only

text: in

text: U

text: the

text: Sp

text: case

text: where

text: the

text: argument

text: contains

text: an

text: additional

text: subtlety.

text: O

text: Proposition

text: (a)

text: The

text: ﬁrst

text: column

text: v

text: of

text: a

text: matrix

text: from

text: (n)

text: is

text: distributed

text: R

text: uniformly

text: on

text: the

text: sphere

text: x

text: n

text: x

text: For

text: k

text: n,

text: the

text: k-th

text: column

text: v

text: is

text: distributed

text: uniformly

text: on

text: the

text: subset

text: of

text: this

text: unit

text: sphere

text: that

text: is

text: orthogonal

text: to

text: k

text: v

text: v

text: k

text: U

text: (b)

text: The

text: ﬁrst

text: column

text: v

text: of

text: a

text: matrix

text: from

text: (n)

text: is

text: distributed

text: uniformly

text: on

text: the

text: C

text: sphere

text: z

text: n

text: z

text: For

text: k

text: n,

text: the

text: k-th

text: column

text: v

text: is

text: distributed

text: k

text: uniformly

text: on

text: the

text: subset

text: of

text: this

text: unit

text: sphere

text: that

text: is

text: orthogonal

text: to

text: v

text: v

text: U

text: k

text: (c)

text: Let

text: Q

text: Sp(n).

text: The

text: ﬁrst

text: column

text: v

text: of

text: C(Q)

text: is

text: distributed

text: uniformly

text: on

text: the

text: C

text: sphere

text: z

text: z

text: The

text: second

text: column

text: v

text: is

text: Zv¯

text: where

text: Z

text: is

text: For

text: k

text: n,

text: the

text: column

text: v

text: is

text: distributed

text: uniformly

text: on

text: the

text: subset

text: of

text: this

text: unit

text: sphere

text: that

text: is

text: orthogonal

text: to

text: v

text: v

text: and

text: v

text: Zv¯

text: U

text: Proof.

text: As

text: noted

text: above,

text: we

text: give

text: details

text: only

text: for

text: the

text: case

text: of

text: Sp(n).

text: Form

text: V

text: with

text: columns

text: v

text: v

text: described

text: by

text: the

text: procedure

text: in

text: the

text: Proposition,

text: U

text: and

text: let

text: us

text: show

text: that

text: V

text: is

text: indeed

text: C(Q)

text: for

text: a

text: Q

text: Sp(n).

text: By

text: the

text: construction,

text: C

text: v

text: v

text: form

text: an

text: orthonormal

text: basis

text: of

text: since

text: v

text: Zv¯

text: holds

text: for

text: any

text: vector

text: v.

text: Thus

text: V

text: is

text: unitary.

text: Moreover,

text: V

text: consists

text: of

text: blocks

text: each

text: of

text: which

text: has

text: the

text: real

text: quaternionic

text: form,

text: see

text: Therefore

text: C

text: belongs

text: to

text: the

text: group

text: (not

text: U

text: ensemble

text: yet)

text: Sp(n).

text: Let

text: us

text: show

text: that

text: for

text: any

text: non-random

text: W

text: chosen

text: from

text: the

text: U

text: group

text: C(

text: Sp(n)),

text: the

text: distribution

text: of

text: V

text: and

text: WV

text: are

text: identical.

text: This

text: will

text: prove

text: that

text: C

text: was

text: generated

text: according

text: to

text: the

text: Haar

text: measure

text: by

text: the

text: uniqueness

text: of

text: the

text: U

text: measure

text: invariant

text: with

text: respect

text: to

text: the

text: group

text: action

text: on

text: Sp(n).

text: Denote

text: the

text: columns

text: of

text: WV

text: by

text: u

text: u

text: First

text: note

text: that

text: u

text: Wv

text: is

text: a

text: C

text: vector

text: uniformly

text: distributed

text: on

text: the

text: unit

text: sphere

text: z

text: z

text: since

text: v

text: is.

text: The

text: second

text: column

text: u

text: Wv

text: WZv¯

text: ZWv

text: Zu¯

text: (we

text: used

text: which

text: agrees

text: with

text: our

text: construction.

text: Now

text: inductively,

text: consider

text: u

text: which

text: is

text: orthogonal

text: to

text: u

text: u

text: Consider

text: any

text: unitary

text: U

text: that

text: maps

text: span

text: u

text: u

text: to

text: itself.

text: Then

text: W

text: maps

text: span

text: v

text: v

text: to

text: itself.

text: By

text: the

text: construction

text: this

text: means

text: that

text: v

text: and

text: W

text: are

text: distributed

text: identically.

text: Thus

text: u

text: ##LTLine##


text: TRUNCATIONS OF UNITARY ENSEMBLES 7 (cid:104) (cid:105) † where v, u = v u. In fact, we have an isometry ı of Hilbert spaces L2(µ) and C Z (cid:96)2( n) or (cid:96)2( ) deﬁned by + (cid:55)→ ∈ Z ı : zm Ume , m . (2.4) 1 Now note that during the Gram-Schmidt procedure above, e was not changed, 1 and therefore our CMV-ﬁcation algorithm can be described as C † U = W W (2.5) † for some unitary W with We = W e = e . This means that not only the C 1 1 1 eigenvalues of U and coincide, but also that the spectral measures (2.3) of U and C with respect to e are identical. In fact, two unitary CMV matrices with identical 1 spectral measures necessarily coincide, see [23, Prop 3.3]. Thus from the physical point of view, one may want to think that given a physical system modeled by a unitary matrix U with a chosen vector e (viewed as the initial 1 state of the system) there is a unique choice of basis in which the matrix attains the “minimal complexity” CMV form. Note however that in the case of systems with time-reversal invariance (COE and CSE) the associated CMV form will not possess the same symmetry properties. In Section , we introduce a variant of the CMV form that does possess the requisite symmetry. Moreover, the matrix used to conjugate to this new canonical form belongs to the othogonal/symplectic group. From the mathematical point of view, one of the advantages of reducing a uni- tary matrix or operator U to the CMV form, apart from the latter being sparse and having the same the spectral measure, is that there is a direct connection to the theory of orthogonal polynomials on the unit circle. Indeed, characteristic polyno- C mials of (α , . . . , α − ), k = 1, 2, . . ., coincide with the orthogonal polynomials Φ 0 k 1 k associated to the spectral measure µ of U with respect to e (see Appendix ): 1 − C det(zI (α , . . . , α − )) = Φ (z). (2.6) k 0 k 1 k In particular, this means that these polynomials satisfy Szeg˝o’s recurrence (B.1). We stress that the recurrence coeﬃcients in (B.1) are precisely the coeﬃcients from the CMV matrix. We will refer to them as the Verblunsky coeﬃcients. We collect all the needed facts from the theory of orthogonal polynomials in Appendix . For more details the reader is referred to the monographs [30, 31]. U For the quaternionic ensembles CSE(n) and Sp(n) we will have to deal with × the 2 2 matrix-valued spectral measures and with block CMV forms which we deﬁne next. × D Recall that µ is an l l matrix-valued probability measure on ∂ if it is a D × countably additive mapping from Borel subsets of ∂ to positive semi-deﬁnite l l D × matrices, with the normalization condition µ(∂ ) = I . The l l spectral measure × l × of an ln ln unitary matrix U with respect to vectors e , e , . . . , e is the l l ≤ ≤ 1 2 l matrix-valued measure dµ with entries dµ (1 j, k l) determined by jk

text: TRUNCATIONS

text: OF

text: UNITARY

text: ENSEMBLES

text: where

text: v,

text: u

text: v

text: u.

text: In

text: fact,

text: we

text: have

text: an

text: isometry

text: ı

text: of

text: Hilbert

text: spaces

text: and

text: C

text: Z

text: n)

text: or

text: deﬁned

text: by

text: Z

text: ı

text: zm

text: Ume

text: m

text: Now

text: note

text: that

text: during

text: the

text: Gram-Schmidt

text: procedure

text: above,

text: e

text: was

text: not

text: changed,

text: and

text: therefore

text: our

text: CMV-ﬁcation

text: algorithm

text: can

text: be

text: described

text: as

text: C

text: U

text: W

text: W

text: for

text: some

text: unitary

text: W

text: with

text: We

text: W

text: e

text: e

text: This

text: means

text: that

text: not

text: only

text: the

text: C

text: eigenvalues

text: of

text: U

text: and

text: coincide,

text: but

text: also

text: that

text: the

text: spectral

text: measures

text: of

text: U

text: and

text: C

text: with

text: respect

text: to

text: e

text: are

text: identical.

text: In

text: fact,

text: two

text: unitary

text: CMV

text: matrices

text: with

text: identical

text: spectral

text: measures

text: necessarily

text: coincide,

text: see

text: Prop

text: Thus

text: from

text: the

text: physical

text: point

text: of

text: view,

text: one

text: may

text: want

text: to

text: think

text: that

text: given

text: a

text: physical

text: system

text: modeled

text: by

text: a

text: unitary

text: matrix

text: U

text: with

text: a

text: chosen

text: vector

text: e

text: (viewed

text: as

text: the

text: initial

text: state

text: of

text: the

text: system)

text: there

text: is

text: a

text: unique

text: choice

text: of

text: basis

text: in

text: which

text: the

text: matrix

text: attains

text: the

text: “minimal

text: complexity”

text: CMV

text: form.

text: Note

text: however

text: that

text: in

text: the

text: case

text: of

text: systems

text: with

text: time-reversal

text: invariance

text: (COE

text: and

text: CSE)

text: the

text: associated

text: CMV

text: form

text: will

text: not

text: possess

text: the

text: same

text: symmetry

text: properties.

text: In

text: Section

text: we

text: introduce

text: a

text: variant

text: of

text: the

text: CMV

text: form

text: that

text: does

text: possess

text: the

text: requisite

text: symmetry.

text: Moreover,

text: the

text: matrix

text: used

text: to

text: conjugate

text: to

text: this

text: new

text: canonical

text: form

text: belongs

text: to

text: the

text: othogonal/symplectic

text: group.

text: From

text: the

text: mathematical

text: point

text: of

text: view,

text: one

text: of

text: the

text: advantages

text: of

text: reducing

text: a

text: uni-

text: tary

text: matrix

text: or

text: operator

text: U

text: to

text: the

text: CMV

text: form,

text: apart

text: from

text: the

text: latter

text: being

text: sparse

text: and

text: having

text: the

text: same

text: the

text: spectral

text: measure,

text: is

text: that

text: there

text: is

text: a

text: direct

text: connection

text: to

text: the

text: theory

text: of

text: orthogonal

text: polynomials

text: on

text: the

text: unit

text: circle.

text: Indeed,

text: characteristic

text: polyno-

text: C

text: mials

text: of

text: (α

text: α

text: k

text: coincide

text: with

text: the

text: orthogonal

text: polynomials

text: Φ

text: k

text: k

text: associated

text: to

text: the

text: spectral

text: measure

text: µ

text: of

text: U

text: with

text: respect

text: to

text: e

text: (see

text: Appendix

text: C

text: det(zI

text: (α

text: α

text: Φ

text: (z).

text: k

text: k

text: k

text: In

text: particular,

text: this

text: means

text: that

text: these

text: polynomials

text: satisfy

text: Szeg˝o’s

text: recurrence

text: We

text: stress

text: that

text: the

text: recurrence

text: coeﬃcients

text: in

text: are

text: precisely

text: the

text: coeﬃcients

text: from

text: the

text: CMV

text: matrix.

text: We

text: will

text: refer

text: to

text: them

text: as

text: the

text: Verblunsky

text: coeﬃcients.

text: We

text: collect

text: all

text: the

text: needed

text: facts

text: from

text: the

text: theory

text: of

text: orthogonal

text: polynomials

text: in

text: Appendix

text: For

text: more

text: details

text: the

text: reader

text: is

text: referred

text: to

text: the

text: monographs

text: U

text: For

text: the

text: quaternionic

text: ensembles

text: CSE(n)

text: and

text: Sp(n)

text: we

text: will

text: have

text: to

text: deal

text: with

text: the

text: matrix-valued

text: spectral

text: measures

text: and

text: with

text: block

text: CMV

text: forms

text: which

text: we

text: deﬁne

text: next.

text: D

text: Recall

text: that

text: µ

text: is

text: an

text: l

text: l

text: matrix-valued

text: probability

text: measure

text: on

text: if

text: it

text: is

text: a

text: D

text: countably

text: additive

text: mapping

text: from

text: Borel

text: subsets

text: of

text: to

text: positive

text: semi-deﬁnite

text: l

text: l

text: D

text: matrices,

text: with

text: the

text: normalization

text: condition

text: µ(∂

text: I

text: The

text: l

text: l

text: spectral

text: measure

text: l

text: of

text: an

text: ln

text: ln

text: unitary

text: matrix

text: U

text: with

text: respect

text: to

text: vectors

text: e

text: e

text: e

text: is

text: the

text: l

text: l

text: l

text: matrix-valued

text: measure

text: dµ

text: with

text: entries

text: dµ

text: j,

text: k

text: l)

text: determined

text: by

text: jk


text: 10 TRUNCATIONS OF UNITARY ENSEMBLES ∈ C ≤ ≤ with α , β , 1 j n. Now Um = (BS)Dm(BS)R and a simple calculation j j (cid:88) implies (cid:104) (cid:105) (cid:104) (cid:105) n | | | | e , C(U)me = e , C(U)me = ( α 2 + β 2)eimθj, 1 1 2 2 j j j=1 (cid:104) (cid:105) (cid:104) (cid:105) e , C(U)me = e , C(U)me = 0. 1 2 2 1 × Therefore the 2 2 spectral measure (2.7) of U is indeed of the form (3.5) with µ = | | | | (cid:81) j α 2 + β 2. By Proposition 2.2(c), the ﬁrst complex column of C((BS)R), that j j C is [α , β , . . . , α , β ]T , is uniformly distributed on the unit sphere of 2n. Then 1 1 n n − by Lemma C.1(e), µ ’s are distributed according to (2n 1)! n µ dµ . . . dµ − j j=1 j 1 n (cid:3)1 on (3.3) as required. Proposition 3.2. SO (a) The spectral measure (cid:88)of (2(cid:0)n) is (cid:1) n 1µ δ + δ − , (3.6) 2 j eiθj e iθj j=1 where (cid:88) ≤ ≤ 0 < θ < π (1 j n), (3.7) j n ≤ ≤ µ = 1, µ > 0 (1 j n), (3.8) j j j=1 and the joint distribution of eiθ1, . . . , eiθn, µ1, . . . , µn−1 is | | × − 2n−11n!πn ∆(2 cos θ1, . . . , 2 cos θn) 2 dθ1 . . . dθn (n 1)!dµ1 . . . dµn−1. O \ SO (b) The spectral (cid:88)measure(cid:0)of (2n) (cid:1) (2n) is − n 1 1µ δ + δ − + µ δ + µ δ− , (3.9) 2 j eiθj e iθj n 1 n+1 1 j=1 where (cid:88) ≤ ≤ − 0 < θ < π (1 j n 1), (3.10) j n+1 ≤ ≤ µ = 1, µ > 0 (1 j n + 1), (3.11) j j j=1 ##LTLine## ##LTLine## ##LTLine##

text: TRUNCATIONS

text: OF

text: UNITARY

text: ENSEMBLES

text: C

text: with

text: α

text: β

text: j

text: n.

text: Now

text: Um

text: (BS)Dm(BS)R

text: and

text: a

text: simple

text: calculation

text: j

text: j

text: implies

text: n

text: e

text: C(U)me

text: e

text: C(U)me

text: α

text: β

text: j

text: j

text: e

text: C(U)me

text: e

text: C(U)me

text: Therefore

text: the

text: spectral

text: measure

text: of

text: U

text: is

text: indeed

text: of

text: the

text: form

text: with

text: µ

text: j

text: α

text: β

text: By

text: Proposition

text: the

text: ﬁrst

text: complex

text: column

text: of

text: C((BS)R),

text: that

text: j

text: j

text: C

text: is

text: [α

text: β

text: α

text: β

text: ]T

text: is

text: uniformly

text: distributed

text: on

text: the

text: unit

text: sphere

text: of

text: Then

text: n

text: n

text: by

text: Lemma

text: µ

text: ’s

text: are

text: distributed

text: according

text: to

text: n

text: µ

text: dµ

text: dµ

text: j

text: j

text: n

text: on

text: as

text: required.

text: Proposition

text: SO

text: (a)

text: The

text: spectral

text: measure

text: is

text: n

text: 1µ

text: δ

text: δ

text: j

text: eiθj

text: e

text: iθj

text: where

text: θ

text: π

text: j

text: n),

text: j

text: n

text: µ

text: µ

text: j

text: n),

text: j

text: j

text: and

text: the

text: joint

text: distribution

text: of

text: eiθn,

text: µn−1

text: is

text: cos

text: cos

text: θn)

text: dθn

text: (n

text: O

text: SO

text: (b)

text: The

text: spectral

text: is

text: n

text: 1µ

text: δ

text: δ

text: µ

text: δ

text: µ

text: δ−

text: j

text: eiθj

text: e

text: iθj

text: n

text: where

text: θ

text: π

text: j

text: n

text: j

text: µ

text: µ

text: j

text: n

text: j

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 2 C. Chappert, A. Fert, and N. Van Dau, Nature Materials 6, 81135 (N2.00O7t)s.uka, Y. Hanawa, and S. Nagakura, Phys. Status Solidi (a) 43, K127 (1977). 3 W. H. Meiklejohn and C. P. Bean, Phys. Rev. Lett. 105, 904 (11695K7.).Suzuki, T. Kaneko, H. Yoshida, Y. Obi, H. Fujimori, and H. Morita, J. Alloys Compd. 306, 66 (2000). 4 J. Nogu´es and I.K. Schuller, J. Magn. Magn. Mater. 192, 203 1(719A99. )L.eineweber, R. Niewa, H. Jacobs, and W. Kockelmann, J. Mater. Chem. 10, 2827 (2000). 5 A. E. Berkowitz and K. Takano, J. Magn. Magn. Mater. 200, 15852K–.57S0uz(u1k99i,9Y).. Yamaguchi, T. Kaneko, H. Yoshida, Y. Obi, H. Fujimori, and H. Morita, J. Phys. Soc. Jpn. 70, 1084 (2001). 6 R. L. Stamps, J. Phys. D. Appl. Phys. 34, 444 (2001). 19 W. Lambrecht, M. Prikhodko, and M. Miao, Phys. Rev. B 68, 174411 (2003). 7 M. Kiwi, J. Magn. Magn. Mater. 234, 584 (2001). 20 H. Yang, H. Al-Brithen, E. Trifan, D. C. Ingram, and A. R. Smith, J. Appl. Phys. 91, 1053 (2002). 8 H. Ohldag, A. Scholl, F. Nolting, E. Arenholz, S. Maat, A. Y2o1unGg.,KM. .WCiallriaeym,saonndaJn.dSWto¨.hHr,.PHhaylls,. ARcetva. MLeettta.ll9.11,, 02127(2109353). (2003). 22 F. Radu and H. Zabel, Magnetic Heterostructures, Exchange Bias Eﬀect of Ferro-/Antiferromagnetic Heterostructures, 9 K. O´Grady, L. E. Fernandez-Outon, and G. Vallejo-Fernandez, JS.pMrinaggenr. TMraagcnts. iMnaMteord. e3r2n2P, h8y8s3ic(s20V1o0l)u.me 227 (Springer Berlin Heidelberg, 2008). 10 J. P. Nozi`eres et al., J. Appl. Phys. 87, 3920 (2000). 23 M. Tsunoda, K. Imakita, M. Naka, and M. Takahashi, J. Magn. Magn. Mater. 304, 55 (2006). 11 M. Ali et al., Phys. Rev. B 68, 214420 (2003). 24 R. A. Reck and D. L. Fry, Phys. Rev. 184, 492 (1969). 12 M. Rickart, A. Guedes, J. Ventura, J. B. Sousa, and P. P. Frei2t5asD, J..MAapuprli., PEh.yKs.ay9,7D, .10SKch1o1l0l, (a2n0d05J).. K. Howard, J. Appl. Phys. 62, 2929 (1987). 13 B. J. Glaister and G. M. Mudd, Minerals Engineering 23, 4382(620M10. )J.. Carey, N. Smith, B. A. Gurney, J. R. Childress, and T. Lin, J. Appl. Phys. 89, 6579 (2001). 14 N. A. Gokcen, Bull. Alloy Phase Diagr. 11, 33 (1990). 27 N.B. Pilling, R. E. Bedworth, J. Inst. Met. 29, 529 (1923).

label: reference

text: C.

text: Chappert,

text: A.

text: Fert,

text: and

text: N.

text: Van

text: Dau,

text: Nature

text: Materials

text: Y.

text: Hanawa,

text: and

text: S.

text: Nagakura,

text: Phys.

text: Status

text: Solidi

text: (a)

text: W.

text: H.

text: Meiklejohn

text: and

text: C.

text: P.

text: Bean,

text: Phys.

text: Rev.

text: Lett.

text: T.

text: Kaneko,

text: H.

text: Yoshida,

text: Y.

text: Obi,

text: H.

text: Fujimori,

text: and

text: H.

text: Morita,

text: J.

text: Alloys

text: Compd.

text: J.

text: Nogu´es

text: and

text: I.K.

text: Schuller,

text: J.

text: Magn.

text: Magn.

text: Mater.

text: )L.eineweber,

text: R.

text: Niewa,

text: H.

text: Jacobs,

text: and

text: W.

text: Kockelmann,

text: J.

text: Mater.

text: Chem.

text: A.

text: E.

text: Berkowitz

text: and

text: K.

text: Takano,

text: J.

text: Magn.

text: Magn.

text: Mater.

text: Yamaguchi,

text: T.

text: Kaneko,

text: H.

text: Yoshida,

text: Y.

text: Obi,

text: H.

text: Fujimori,

text: and

text: H.

text: Morita,

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: R.

text: L.

text: Stamps,

text: J.

text: Phys.

text: D.

text: Appl.

text: Phys.

text: W.

text: Lambrecht,

text: M.

text: Prikhodko,

text: and

text: M.

text: Miao,

text: Phys.

text: Rev.

text: B

text: M.

text: Kiwi,

text: J.

text: Magn.

text: Magn.

text: Mater.

text: H.

text: Yang,

text: H.

text: Al-Brithen,

text: E.

text: Trifan,

text: D.

text: C.

text: Ingram,

text: and

text: A.

text: R.

text: Smith,

text: J.

text: Appl.

text: Phys.

text: H.

text: Ohldag,

text: A.

text: Scholl,

text: F.

text: Nolting,

text: E.

text: Arenholz,

text: S.

text: Maat,

text: A.

text: ARcetva.

text: F.

text: Radu

text: and

text: H.

text: Zabel,

text: Magnetic

text: Heterostructures,

text: Exchange

text: Bias

text: Eﬀect

text: of

text: Ferro-/Antiferromagnetic

text: Heterostructures,

text: K.

text: O´Grady,

text: L.

text: E.

text: Fernandez-Outon,

text: and

text: G.

text: Vallejo-Fernandez,

text: JS.pMrinaggenr.

text: TMraagcnts.

text: iMnaMteord.

text: (Springer

text: Berlin

text: Heidelberg,

text: J.

text: P.

text: Nozi`eres

text: et

text: al.,

text: J.

text: Appl.

text: Phys.

text: M.

text: Tsunoda,

text: K.

text: Imakita,

text: M.

text: Naka,

text: and

text: M.

text: Takahashi,

text: J.

text: Magn.

text: Magn.

text: Mater.

text: M.

text: Ali

text: et

text: al.,

text: Phys.

text: Rev.

text: B

text: R.

text: A.

text: Reck

text: and

text: D.

text: L.

text: Fry,

text: Phys.

text: Rev.

text: M.

text: Rickart,

text: A.

text: Guedes,

text: J.

text: Ventura,

text: J.

text: B.

text: Sousa,

text: and

text: P.

text: P.

text: J..MAapuprli.,

text: K.

text: Howard,

text: J.

text: Appl.

text: Phys.

text: B.

text: J.

text: Glaister

text: and

text: G.

text: M.

text: Mudd,

text: Minerals

text: Engineering

text: )J..

text: Carey,

text: N.

text: Smith,

text: B.

text: A.

text: Gurney,

text: J.

text: R.

text: Childress,

text: and

text: T.

text: Lin,

text: J.

text: Appl.

text: Phys.

text: N.

text: A.

text: Gokcen,

text: Bull.

text: Alloy

text: Phase

text: Diagr.

text: N.B.

text: Pilling,

text: R.

text: E.

text: Bedworth,

text: J.

text: Inst.

text: Met.


text: 16 A.-Y. Zhou [29] Kurtz, D. W., Hambleton, K. M., Shibahashi, H., et al. 2015, MNRAS, 446, 1223 Validation of the frequency modulation technique applied to the pulsating delta Sct–gamma Dor eclipsing binary star KIC 8569819 [30] Le Contel, J.-M., Mathias, P., Chapellier, E., & Valtier, J.-C. 2001, A&A, 380, 277: 53 Psc [31] Lutz, R., et al. 2008, in Hot Subdwarf Stars and Related Objects, eds. U. Heber, S. Jeffery, & R. Napiwotzki, ASP Conf., 392, p.339 [32] Lutz, R., Schuh, S., Silvotti, R., et al. 2009, A&A, 496, 469: V391 Pegasi [33] Maceroni, C., Lehmann, H., Da Silva, R., et al. 2014, A&A, 563, 59 (arXiv:1401.3130): KIC 3858884, hybrid DSCT KIC 3858884: a hybrid delta Scuti pulsator in a highly eccentric eclipsing binary. [34] Miglio, A, Montalba´n., J., & Dupret, M.-A. 2007, MNRAS, 375, L21 [35] Pamyatnykh, A. A. 1999, Acta Astron. 49, 119 [36] Pandey, C. P., et al. 2011, CoAst, 162, 21 [37] Reed, M. D., Kawaler, S. D., Østensen, R. H., et al. 2010, MNRAS, 409, 1496 [38] Ripepi, V. et al. 2011, MNRAS, 416, 1535: CoRoT 102699796 [39] Rowe, J. F., Matthews, J. M., Cameron, C., et al. 2006, CoAst, 148, 34 Discovery of hybrid gamma Dor and delta Sct pulsations in BD+18 4914 through MOST spacebased pho- tometry [40] Samus, N. N., Durlevich, O. V., Kazarovets, E. V., Kireeva, N. N., Pastukhova, E. N., Zharova, A. V., et al., General Catalogue of Variable Stars: http://www.sai.msu.su/gcvs/gcvs/index.htm [41] Schuh, S., Huber, J., Dreizler, S., et al. 2006, A&A, 445, L31. [42] Seaton, M. J. 2005 MNRAS, 362, L1 [43] Silvotti, R., et al. 2007, Nature, 449, 189 [44] So´dor, A´ ., De Cat, P., Wright, D. J., et al. 2014, MNRAS, 438, 3535 (arXiv:1312.6307): HD 25558: SB2, SPB+SPB [45] Tkachenko, A., et al., 2013, A&A, 556, 52: Detection of a large sample of gamma Doradus stars from Kepler space photometry and high-resolution ground-based spectroscopy [46] Vauclair, G., Solheim, J.-E., & Østensen, R. H. 2005, A&A, 433, 1097: Abell 43 [47] Zdravkov, T., & Pamyatnykh A. A. 2008, CoAst, 157, 385 (arXiv:0810.1609) [48] Zhang, X. B., Deng, L. C., Luo, C. Q. 2012, AJ, 144, 141: A Probable Hybrid gamma Dor-delta Scuti Variable Discovered in the Open Cluster NGC 2126 [49] Zhou, A.-Y., 2010, arXiv:1002.2729v6 (last revised 9 Dec 2014)

text: A.-Y.

text: Zhou

text: Kurtz,

text: D.

text: W.,

text: Hambleton,

text: K.

text: M.,

text: Shibahashi,

text: H.,

text: et

text: al.

text: MNRAS,

text: Validation

text: of

text: the

text: frequency

text: modulation

text: technique

text: applied

text: to

text: the

text: pulsating

text: delta

text: Sct–gamma

text: Dor

text: eclipsing

text: binary

text: star

text: KIC

text: Le

text: Contel,

text: J.-M.,

text: Mathias,

text: P.,

text: Chapellier,

text: E.,

text: Valtier,

text: J.-C.

text: A&A,

text: Psc

text: Lutz,

text: R.,

text: et

text: al.

text: in

text: Hot

text: Subdwarf

text: Stars

text: and

text: Related

text: Objects,

text: eds.

text: U.

text: Heber,

text: S.

text: Jeffery,

text: R.

text: Napiwotzki,

text: ASP

text: Conf.,

text: Lutz,

text: R.,

text: Schuh,

text: S.,

text: Silvotti,

text: R.,

text: et

text: al.

text: A&A,

text: Pegasi

text: Maceroni,

text: C.,

text: Lehmann,

text: H.,

text: Da

text: Silva,

text: R.,

text: et

text: al.

text: A&A,

text: KIC

text: hybrid

text: DSCT

text: KIC

text: a

text: hybrid

text: delta

text: Scuti

text: pulsator

text: in

text: a

text: highly

text: eccentric

text: eclipsing

text: binary.

text: Miglio,

text: A,

text: Montalba´n.,

text: J.,

text: Dupret,

text: M.-A.

text: MNRAS,

text: Pamyatnykh,

text: A.

text: A.

text: Acta

text: Astron.

text: Pandey,

text: C.

text: P.,

text: et

text: al.

text: CoAst,

text: Reed,

text: M.

text: D.,

text: Kawaler,

text: S.

text: D.,

text: Østensen,

text: R.

text: H.,

text: et

text: al.

text: MNRAS,

text: Ripepi,

text: V.

text: et

text: al.

text: MNRAS,

text: CoRoT

text: Rowe,

text: J.

text: F.,

text: Matthews,

text: J.

text: M.,

text: Cameron,

text: C.,

text: et

text: al.

text: CoAst,

text: Discovery

text: of

text: hybrid

text: gamma

text: Dor

text: and

text: delta

text: Sct

text: pulsations

text: in

text: through

text: MOST

text: spacebased

text: pho-

text: tometry

text: Samus,

text: N.

text: N.,

text: Durlevich,

text: O.

text: V.,

text: Kazarovets,

text: E.

text: V.,

text: Kireeva,

text: N.

text: N.,

text: Pastukhova,

text: E.

text: N.,

text: Zharova,

text: A.

text: V.,

text: et

text: al.,

text: General

text: Catalogue

text: of

text: Variable

text: Stars:

text: Schuh,

text: S.,

text: Huber,

text: J.,

text: Dreizler,

text: S.,

text: et

text: al.

text: A&A,

text: Seaton,

text: M.

text: J.

text: MNRAS,

text: Silvotti,

text: R.,

text: et

text: al.

text: Nature,

text: So´dor,

text: A´

text: De

text: Cat,

text: P.,

text: Wright,

text: D.

text: J.,

text: et

text: al.

text: MNRAS,

text: HD

text: SPB+SPB

text: Tkachenko,

text: A.,

text: et

text: al.,

text: A&A,

text: Detection

text: of

text: a

text: large

text: sample

text: of

text: gamma

text: Doradus

text: stars

text: from

text: Kepler

text: space

text: photometry

text: and

text: high-resolution

text: ground-based

text: spectroscopy

text: Vauclair,

text: G.,

text: Solheim,

text: J.-E.,

text: Østensen,

text: R.

text: H.

text: A&A,

text: Abell

text: Zdravkov,

text: T.,

text: Pamyatnykh

text: A.

text: A.

text: CoAst,

text: Zhang,

text: X.

text: B.,

text: Deng,

text: L.

text: C.,

text: Luo,

text: C.

text: Q.

text: AJ,

text: A

text: Probable

text: Hybrid

text: gamma

text: Dor-delta

text: Scuti

text: Variable

text: Discovered

text: in

text: the

text: Open

text: Cluster

text: NGC

text: Zhou,

text: A.-Y.,

text: (last

text: revised

text: Dec


text: 6 A.-Y. Zhou same area with δ Sct stars and is not conﬁned to the overlapping region in the H-R diagram. They behave differently in the g-mode regime from γ Dor stars. They likely form a different class of pulsating stars and are not merely a simple mixture of δ Sct and γ Dor types pulsations. Thus a different driving mechanism is demanded to excite the observed g-modes of hybrid stars. A detailed discussion on δ Sct–γ Dor hybrids can be found in Grigahce`ne et al.(2010) and Hareter (2013). (iv) Modiﬁed modelling is demanded to explain the existence of hybrid pulsations. Attempts towards seismic modelling of the observed hybrid pulsation frequencies, for the β Cep–SPB, δ Sct–γ Dor, and sdBV are rs overwhelming. 4 ENDING REMARKS The goal of this survey is to have an overall view of various hybrid pulsators, actually the variables with multiple variability types. We have extracted the hybrid information based on the Simbad astronomical database. This report presents our preliminary results, the cross table Table 2 along with several related statistics (Tables 3– 4). A web page (http://www.chjaa.org/COB/hybrid/) that allows one to retrieve the hybrid stars of any two types of variability is developed. We draw the readers’ attention to several kinds of hybrid pulsators, including pulsating stars in binary systems, pulsating stars hosting transiting exoplanets, and so on. In the next step, we plan to update the table by checking literature and other database. In the mean time, we are going to do observations on some of the interesting hybrid pulsators. Acknowledgements This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. The author has consulted the GCVS for deﬁnitions and classiﬁcations of variable stars. This research was funded by the National Natural Science Foundation of China (NSFC) under the project No. 11373037.

text: A.-Y.

text: Zhou

text: same

label: list

text: area

text: with

text: δ

text: Sct

text: stars

text: and

text: is

text: not

text: conﬁned

text: to

text: the

text: overlapping

text: region

text: in

text: the

text: H-R

text: diagram.

text: They

text: behave

text: differently

text: in

text: the

text: g-mode

text: regime

text: from

text: γ

text: Dor

text: stars.

text: They

text: likely

text: form

text: a

text: different

text: class

text: of

text: pulsating

text: stars

text: and

text: are

text: not

text: merely

text: a

text: simple

text: mixture

text: of

text: δ

text: Sct

text: and

text: γ

text: Dor

text: types

text: pulsations.

text: Thus

text: a

text: different

text: driving

text: mechanism

text: is

text: demanded

text: to

text: excite

text: the

text: observed

text: g-modes

text: of

text: hybrid

text: stars.

text: A

text: detailed

text: discussion

text: on

text: δ

text: Sct–γ

text: Dor

text: hybrids

text: can

text: be

text: found

text: in

text: Grigahce`ne

text: et

text: and

text: Hareter

text: (iv)

text: Modiﬁed

text: modelling

text: is

text: demanded

text: to

text: explain

text: the

text: existence

text: of

text: hybrid

text: pulsations.

text: Attempts

text: towards

text: seismic

text: modelling

text: of

text: the

text: observed

text: hybrid

text: pulsation

text: frequencies,

text: for

text: the

text: β

text: Cep–SPB,

text: δ

text: Sct–γ

text: Dor,

text: and

text: sdBV

text: are

text: rs

text: overwhelming.

text: ENDING

text: REMARKS

text: The

text: goal

text: of

text: this

text: survey

text: is

text: to

text: have

text: an

text: overall

text: view

text: of

text: various

text: hybrid

text: pulsators,

text: actually

text: the

text: variables

text: with

text: multiple

text: variability

text: types.

text: We

text: have

text: extracted

text: the

text: hybrid

text: information

text: based

text: on

text: the

text: Simbad

text: astronomical

text: database.

text: This

text: report

text: presents

text: our

text: preliminary

text: results,

text: the

text: cross

text: table

text: Table

text: along

text: with

text: several

text: related

text: statistics

text: (Tables

text: 3–

text: A

text: web

text: page

text: that

text: allows

text: one

text: to

text: retrieve

text: the

text: hybrid

text: stars

text: of

text: any

text: two

text: types

text: of

text: variability

text: is

text: developed.

text: We

text: draw

text: the

text: readers’

text: attention

text: to

text: several

text: kinds

text: of

text: hybrid

text: pulsators,

text: including

text: pulsating

text: stars

text: in

text: binary

text: systems,

text: pulsating

text: stars

text: hosting

text: transiting

text: exoplanets,

text: and

text: so

text: on.

text: In

text: the

text: next

text: step,

text: we

text: plan

text: to

text: update

text: the

text: table

text: by

text: checking

text: literature

text: and

text: other

text: database.

text: In

text: the

text: mean

text: time,

text: we

text: are

text: going

text: to

text: do

text: observations

text: on

text: some

text: of

text: the

text: interesting

text: hybrid

text: pulsators.

text: Acknowledgements

text: This

text: research

text: has

text: made

text: use

text: of

text: the

text: SIMBAD

text: database,

text: operated

text: at

text: CDS,

text: Strasbourg,

text: France.

text: The

text: author

text: has

text: consulted

text: the

text: GCVS

text: for

text: deﬁnitions

text: and

text: classiﬁcations

text: of

text: variable

text: stars.

text: This

text: research

text: was

text: funded

text: by

text: the

text: National

text: Natural

text: Science

text: Foundation

text: of

text: China

text: (NSFC)

text: under

text: the

text: project

text: No.


text: Search for the U boson in the process − → → − e+e Uγ, U e+e with the KLOE detector Anthony Palladino Laboratori Nazionali di Frascati, Frascati, Italy on behalf of the KLOE-2 collaboration Dark Matter and Dark Energy are two of the most fundamental open questions in physics today. The existence of a light dark-force mediator has been hypothesized as a possible explanation for several unexplained physical phenomena. A new search for this mediator, the dark photon U, is underway using data collected with the KLOE detector at DAΦNE. We describe the strategy we will use in our search for a resonant peak in the − → electron-positron invariant mass spectrum from the process e+e Uγ → − with U e+e . So far we found no evidence for the process and set a preliminary upper limit on the level of mixing between the secluded dark sector and the standard model. DOI: 10.5506/APhysPolB.46.1001 PACS numbers: 13.66.Hk, 14.80.-j, 12.60.Cn, 95.35.+d 1. Introduction A series of unexpected astrophysical observations have failed to ﬁnd explanations in terms of standard astrophysical or particle physics mod- els [1–10]. Each of these anomalies can be explained, however, if there exists a dark weakly interacting massive particle, WIMP, belonging to a secluded gauge sector [11–15]. A dark vector boson, U, an abelian gauge ﬁeld, may couple the secluded sector to the Standard Model through its ki- netic mixing with the Standard Model electroweak hypercharge gauge ﬁeld, L − = ε2/ F EWF ij . The kinetic mixing parameter, ε, is expected to mix 2 ij − Dark− O be of the order 10 4–10 2 which allows for observable eﬀects in (GeV)- − energy e+e colliders [18–20]. The U boson might be produced in such → collider experiments via several processes: V PU decays, where V and ∗ Submitted to Acta Physica Polonica B on 24 October 2014. (1) ##LTLine##

text: Search

label: title

text: for

text: the

text: U

text: boson

text: in

text: the

text: process

text: e+e

text: Uγ,

text: U

text: e+e

text: with

text: the

text: KLOE

text: detector

text: Anthony

label: author

text: Palladino

text: Laboratori

text: Nazionali

text: di

text: Frascati,

text: Frascati,

text: Italy

text: on

text: behalf

text: of

text: the

text: collaboration

text: Dark

label: abstract

text: Matter

text: and

text: Dark

text: Energy

text: are

text: two

text: of

text: the

text: most

text: fundamental

text: open

text: questions

text: in

text: physics

text: today.

text: The

text: existence

text: of

text: a

text: light

text: dark-force

text: mediator

text: has

text: been

text: hypothesized

text: as

text: a

text: possible

text: explanation

text: for

text: several

text: unexplained

text: physical

text: phenomena.

text: A

text: new

text: search

text: for

text: this

text: mediator,

text: the

text: dark

text: photon

text: U,

text: is

text: underway

text: using

text: data

text: collected

text: with

text: the

text: KLOE

text: detector

text: at

text: DAΦNE.

text: We

text: describe

text: the

text: strategy

text: we

text: will

text: use

text: in

text: our

text: search

text: for

text: a

text: resonant

text: peak

text: in

text: the

text: electron-positron

text: invariant

text: mass

text: spectrum

text: from

text: the

text: process

text: e+e

text: Uγ

text: with

text: U

text: e+e

text: So

text: far

text: we

text: found

text: no

text: evidence

text: for

text: the

text: process

text: and

text: set

text: a

text: preliminary

text: upper

text: limit

text: on

text: the

text: level

text: of

text: mixing

text: between

text: the

text: secluded

text: dark

text: sector

text: and

text: the

text: standard

text: model.

text: DOI:

text: PACS

text: numbers:

text: Introduction

text: A

text: series

text: of

text: unexpected

text: astrophysical

text: observations

text: have

text: failed

text: to

text: ﬁnd

text: explanations

text: in

text: terms

text: of

text: standard

text: astrophysical

text: or

text: particle

text: physics

text: mod-

text: els

text: Each

text: of

text: these

text: anomalies

text: can

text: be

text: explained,

text: however,

text: if

text: there

text: exists

text: a

text: dark

text: weakly

text: interacting

text: massive

text: particle,

text: WIMP,

text: belonging

text: to

text: a

text: secluded

text: gauge

text: sector

text: A

text: dark

text: vector

text: boson,

text: U,

text: an

text: abelian

text: gauge

text: ﬁeld,

text: may

text: couple

text: the

text: secluded

text: sector

text: to

text: the

text: Standard

text: Model

text: through

text: its

text: ki-

text: netic

text: mixing

text: with

text: the

text: Standard

text: Model

text: electroweak

text: hypercharge

text: gauge

text: ﬁeld,

text: L

text: F

text: EWF

text: ij

text: The

text: kinetic

text: mixing

text: parameter,

text: ε,

text: is

text: expected

text: to

text: mix

text: ij

text: Dark−

text: O

text: be

text: of

text: the

text: order

text: which

text: allows

text: for

text: observable

text: eﬀects

text: in

text: (GeV)-

text: energy

text: e+e

text: colliders

text: The

text: U

text: boson

text: might

text: be

text: produced

text: in

text: such

text: collider

text: experiments

text: via

text: several

text: processes:

text: V

text: PU

text: decays,

text: where

text: V

text: and

text: Submitted

text: to

text: Acta

text: Physica

text: Polonica

text: B

text: on

text: October

text: ##LTLine##


text: palladino˙APPB printed on October 16, 2014 3 where we have electrons as our ﬁnal-state leptons ((cid:96) = e) and Γ = → − → − → Total Γ(U e+e ) + Γ(U µ+µ ) + Γ(U hadrons) is the total width. A new KLOE-2 analysis is underway which proposes to search for U bo- − → → − son production in the process e+e Uγ, U e+e . The 3 ﬁnal-state particles of this process are the same as radiative Bhabha scattering. The distinct feature we are searching for is a Breit-Wigner resonant production − peak (at the U boson mass) in the invariant-mass distribution of the e+e − pair. To search for a U boson produced at a ﬁxed-energy e+e collider we use initial-state radiation (ISR) to reduce the center of mass energy and thereby scan the range of possible U boson masses down to 2m . The pro- e cess consists of ﬁnite-width eﬀects for s-channel annihilation subprocesses, non-resonant t-channel U boson exchange, and s-t interference contribu- tions. The ﬁnite-width eﬀects are order Γ /m on the integrated cross U U section so are much smaller than any potential resonance we would observe, but they are critical from a phenomonological perspective and are properly taken into account in the Monte Carlo simulation [31]. The non-resonant t-channel eﬀects would not produce the Breit-Wigner peak in the invariant mass distribution but could, in principle, show up in analyses of angular distributions or asymmetries. The KLOE-2 analysis will focus exclusively on resonant s-channel U boson production. − Using about 1.5 fb 1 of KLOE data collected during 2004–2005 we will search for U boson production in a sample of radiative Bhabha scatter- ing events. The strategy is to select events with the ﬁnal-state electron, ◦ ◦ positron, and photon, all emitted at large angle (55 < θ < 125 ) with re- spect to the beam axis, such that they are explicitly detected in the barrel of the calorimeter. The m variable, computed using energy and momen- track tum conservation, with the assumption of equal-mass oppositely-charged particles, will be used to separate electrons from the more massive muons and pions. We will use Monte Carlo (MC) simulations to estimate the level of back- − → − ground contamination due to the following processes: e+e µ+µ γ, − → − − → − e+e π+π γ, e+e γγ (where one photon converts into an e+e − → → → − pair), and e+e φ ρπ0 π+π π0, as well as other φ decays. Due to the KLOE detector’s excellent eﬃciency at detecting electrons and distin- guishing them from heavier charged particles, we estimate that the sum of all background processes is typically less than 1% in the m distribution. ee None of the background shapes are peaked, eliminating the possiblility of a background mimicking the resonant U boson signal. (cid:113) Several Monte Carlo event generators for radiative Bhabha scattering fail to accurately reproduce the physics at the dielectron mass threshold − due to numerical instabilities in integrations of the form 1 1 4m2 . Due q2 q2 ##LTLine## ##LTLine## ##LTLine##

text: palladino˙APPB

text: printed

text: on

text: October

text: where

text: we

text: have

text: electrons

text: as

text: our

text: ﬁnal-state

text: leptons

text: e)

text: and

text: Γ

text: Total

text: Γ(U

text: e+e

text: Γ(U

text: µ+µ

text: Γ(U

text: hadrons)

text: is

text: the

text: total

text: width.

text: A

text: new

text: analysis

text: is

text: underway

text: which

text: proposes

text: to

text: search

text: for

text: U

text: bo-

text: son

text: production

text: in

text: the

text: process

text: e+e

text: Uγ,

text: U

text: e+e

text: The

text: ﬁnal-state

text: particles

text: of

text: this

text: process

text: are

text: the

text: same

text: as

text: radiative

text: Bhabha

text: scattering.

text: The

text: distinct

text: feature

text: we

text: are

text: searching

text: for

text: is

text: a

text: Breit-Wigner

text: resonant

text: production

text: peak

text: (at

text: the

text: U

text: boson

text: mass)

text: in

text: the

text: invariant-mass

text: distribution

text: of

text: the

text: e+e

text: pair.

text: To

text: search

text: for

text: a

text: U

text: boson

text: produced

text: at

text: a

text: ﬁxed-energy

text: e+e

text: collider

text: we

text: use

text: initial-state

text: radiation

text: (ISR)

text: to

text: reduce

text: the

text: center

text: of

text: mass

text: energy

text: and

text: thereby

text: scan

text: the

text: range

text: of

text: possible

text: U

text: boson

text: masses

text: down

text: to

text: The

text: pro-

text: e

text: cess

text: consists

text: of

text: ﬁnite-width

text: eﬀects

text: for

text: s-channel

text: annihilation

text: subprocesses,

text: non-resonant

text: t-channel

text: U

text: boson

text: exchange,

text: and

text: s-t

text: interference

text: contribu-

text: tions.

text: The

text: ﬁnite-width

text: eﬀects

text: are

text: order

text: Γ

text: /m

text: on

text: the

text: integrated

text: cross

text: U

text: U

text: section

text: so

text: are

text: much

text: smaller

text: than

text: any

text: potential

text: resonance

text: we

text: would

text: observe,

text: but

text: they

text: are

text: critical

text: from

text: a

text: phenomonological

text: perspective

text: and

text: are

text: properly

text: taken

text: into

text: account

text: in

text: the

text: Monte

text: Carlo

text: simulation

text: The

text: non-resonant

text: t-channel

text: eﬀects

text: would

text: not

text: produce

text: the

text: Breit-Wigner

text: peak

text: in

text: the

text: invariant

text: mass

text: distribution

text: but

text: could,

text: in

text: principle,

text: show

text: up

text: in

text: analyses

text: of

text: angular

text: distributions

text: or

text: asymmetries.

text: The

text: analysis

text: will

text: focus

text: exclusively

text: on

text: resonant

text: s-channel

text: U

text: boson

text: production.

text: Using

text: about

text: fb

text: of

text: KLOE

text: data

text: collected

text: during

text: we

text: will

text: search

text: for

text: U

text: boson

text: production

text: in

text: a

text: sample

text: of

text: radiative

text: Bhabha

text: scatter-

text: ing

text: events.

text: The

text: strategy

text: is

text: to

text: select

text: events

text: with

text: the

text: ﬁnal-state

text: electron,

text: positron,

text: and

text: photon,

text: all

text: emitted

text: at

text: large

text: angle

text: θ

text: with

text: re-

text: spect

text: to

text: the

text: beam

text: axis,

text: such

text: that

text: they

text: are

text: explicitly

text: detected

text: in

text: the

text: barrel

text: of

text: the

text: calorimeter.

text: The

text: m

text: variable,

text: computed

text: using

text: energy

text: and

text: momen-

text: track

text: tum

text: conservation,

text: with

text: the

text: assumption

text: of

text: equal-mass

text: oppositely-charged

text: particles,

text: will

text: be

text: used

text: to

text: separate

text: electrons

text: from

text: the

text: more

text: massive

text: muons

text: and

text: pions.

text: We

text: will

text: use

text: Monte

text: Carlo

text: (MC)

text: simulations

text: to

text: estimate

text: the

text: level

text: of

text: back-

text: ground

text: contamination

text: due

text: to

text: the

text: following

text: processes:

text: e+e

text: µ+µ

text: γ,

text: e+e

text: π+π

text: γ,

text: e+e

text: γγ

text: (where

text: one

text: photon

text: converts

text: into

text: an

text: e+e

text: pair),

text: and

text: e+e

text: φ

text: ρπ0

text: π+π

text: as

text: well

text: as

text: other

text: φ

text: decays.

text: Due

text: to

text: the

text: KLOE

text: detector’s

text: excellent

text: eﬃciency

text: at

text: detecting

text: electrons

text: and

text: distin-

text: guishing

text: them

text: from

text: heavier

text: charged

text: particles,

text: we

text: estimate

text: that

text: the

text: sum

text: of

text: all

text: background

text: processes

text: is

text: typically

text: less

text: than

text: in

text: the

text: m

text: distribution.

text: ee

text: of

text: the

text: background

text: shapes

text: are

text: peaked,

text: eliminating

text: the

text: possiblility

text: of

text: a

text: background

text: mimicking

text: the

text: resonant

text: U

text: boson

text: signal.

text: Several

text: Monte

text: Carlo

text: event

text: generators

text: for

text: radiative

text: Bhabha

text: scattering

text: fail

text: to

text: accurately

text: reproduce

text: the

text: physics

text: at

text: the

text: dielectron

text: mass

text: threshold

text: due

text: to

text: numerical

text: instabilities

text: in

text: integrations

text: of

text: the

text: form

text: Due

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: palladino˙APPB printed on October 16, 2014 7 [31] L. Barz`e, et al., Eur. Phys. J. C, 71 (2011), 1680 [32] G. C. Feldman, R. D. Cousins, Physical Rev. D 57, 3873 (1998) [33] D. Babusci, et al. (KLOE-2 Collaboration), Phys. Lett. B 736 (2014), 459–464 [34] H. Czyz˙, et al., Eur. Phys. J. C, 39 411 (2005) [35] E. M. Riordan, et al. (E141 Collaboration), Phys. Rev. Lett. 59 (1987), 755 [36] A. Bross, et al. (E774 Collaboration), Phys. Rev. Lett. 67 (1991), 2942 [37] S. Abrahamyan, et al. (APEX Collab.), Phys. Rev. Lett. 107 (2011), 191804 [38] P. Adlarson, et al. (WASA-at-COSY Collab.), Phys. Lett. B 726 (2013), 187 [39] G. Agakishiev, et al. (HADES Collab.) Phys. Lett. B 731, (2014), 265–271 [40] H. Merkel, et al. (A1 Collaboration), Phys. Rev. Lett. 112 (2014), 221802 [41] J. P. Lees, et al. (BaBar Collaboration), [arXiv:1406.2980] (2014) [42] G. Amelino-Camelia, et al., Eur. Phys. J. C, 68, 619 (2010) [43] A. Balla et al., JINST 9, C01014 (2014).

text: palladino˙APPB

text: printed

text: on

text: October

text: L.

text: Barz`e,

text: et

text: al.,

text: Eur.

text: Phys.

text: J.

text: C,

text: G.

text: C.

text: Feldman,

text: R.

text: D.

text: Cousins,

text: Physical

text: Rev.

text: D

text: D.

text: Babusci,

text: et

text: al.

text: Collaboration),

text: Phys.

text: Lett.

text: B

text: H.

text: Czyz˙,

text: et

text: al.,

text: Eur.

text: Phys.

text: J.

text: C,

text: E.

text: M.

text: Riordan,

text: et

text: al.

text: Collaboration),

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Bross,

text: et

text: al.

text: Collaboration),

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Abrahamyan,

text: et

text: al.

text: (APEX

text: Collab.),

text: Phys.

text: Rev.

text: Lett.

text: P.

text: Adlarson,

text: et

text: al.

text: (WASA-at-COSY

text: Collab.),

text: Phys.

text: Lett.

text: B

text: G.

text: Agakishiev,

text: et

text: al.

text: (HADES

text: Collab.)

text: Phys.

text: Lett.

text: B

text: H.

text: Merkel,

text: et

text: al.

text: Collaboration),

text: Phys.

text: Rev.

text: Lett.

text: J.

text: P.

text: Lees,

text: et

text: al.

text: (BaBar

text: Collaboration),

text: G.

text: Amelino-Camelia,

text: et

text: al.,

text: Eur.

text: Phys.

text: J.

text: C,

text: A.

text: Balla

text: et

text: al.,

text: JINST


text: x 10−3 T = 0.5 2.5 T = 1 T = 3 T = 5 T = 10 2 1.5 1 0.5 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 Figure 11: Bin size dependence in the U-pattern volatilities: Time average of the cross-sectional volatility for the S&P 500 for 5 diﬀerent values of bin size T . intra-day patterns in relative prices in order to characterize “atypical days” for indexes and “anomalous behaviours” for stocks. The time average of the cross-sectional moments represents the average behaviour of a particular index moment during an average day. In ﬁgure 8 we showed how each path represents the evolution of a particular index moment for one of the days of the period under analysis (i.e. one path, one day moment). If we look directly into the prices of the CAC 40 and S&P 500, we can observe during day 11 a fall of the prices of the stocks that compose both indexes. During the days before and following day 11, the (index) moments move along our intra-day pattern. Moreover, if we pick randomly one day from our period of analysis, in most of the cases our index during that day will behave as our intra-day seasonality (as in ﬁgure 13), but the one for day 11 will not. In ﬁgure 13 we show our (cross-sectional) intra-day seasonalities for the mean (a) and volatility (b) in blue and in clear blue the respective cross- sectional stock moments for 3 days randomly picked. The average behaviour (of the moments) of our index during these days moves along with our intra- day pattern. This is not the case of the curve corresponding to the day 11 shown in red which clearly diverges from the expected behaviour. This is 14 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: x

font: default

text: T

text: T

text: T

text: T

text: T

text: Figure

text: Bin

label: caption

text: size

text: dependence

text: in

text: the

text: U-pattern

text: volatilities:

text: Time

text: average

text: of

text: the

text: cross-sectional

text: volatility

text: for

text: the

text: S&P

text: for

text: diﬀerent

text: values

text: of

text: bin

text: size

text: T

text: intra-day

text: patterns

text: in

text: relative

text: prices

text: in

text: order

text: to

text: characterize

text: “atypical

text: days”

text: for

text: indexes

text: and

text: “anomalous

text: behaviours”

text: for

text: stocks.

text: The

text: time

text: average

text: of

text: the

text: cross-sectional

text: moments

text: represents

text: the

text: average

text: behaviour

text: of

text: a

text: particular

text: index

text: moment

text: during

text: an

text: average

text: day.

text: In

text: ﬁgure

text: we

text: showed

text: how

text: each

text: path

text: represents

text: the

text: evolution

text: of

text: a

text: particular

text: index

text: moment

text: for

text: one

text: of

text: the

text: days

text: of

text: the

text: period

text: under

text: analysis

text: (i.e.

text: one

text: path,

text: one

text: day

text: moment).

text: If

text: we

text: look

text: directly

text: into

text: the

text: prices

text: of

text: the

text: CAC

text: and

text: S&P

text: we

text: can

text: observe

text: during

text: day

text: a

text: fall

text: of

text: the

text: prices

text: of

text: the

text: stocks

text: that

text: compose

text: both

text: indexes.

text: During

text: the

text: days

text: before

text: and

text: following

text: day

text: the

text: (index)

text: moments

text: move

text: along

text: our

text: intra-day

text: pattern.

text: Moreover,

text: if

text: we

text: pick

text: randomly

text: one

text: day

text: from

text: our

text: period

text: of

text: analysis,

text: in

text: most

text: of

text: the

text: cases

text: our

text: index

text: during

text: that

text: day

text: will

text: behave

text: as

text: our

text: intra-day

text: seasonality

text: (as

text: in

text: ﬁgure

text: but

text: the

text: one

text: for

text: day

text: will

text: not.

text: In

text: ﬁgure

text: we

text: show

text: our

text: (cross-sectional)

text: intra-day

text: seasonalities

text: for

text: the

text: mean

text: (a)

text: and

text: volatility

text: (b)

text: in

text: blue

text: and

text: in

text: clear

text: blue

text: the

text: respective

text: cross-

text: sectional

text: stock

text: moments

text: for

text: days

text: randomly

text: picked.

text: The

text: average

text: behaviour

text: (of

text: the

text: moments)

text: of

text: our

text: index

text: during

text: these

text: days

text: moves

text: along

text: with

text: our

text: intra-

text: day

text: pattern.

text: This

text: is

text: not

text: the

text: case

text: of

text: the

text: curve

text: corresponding

text: to

text: the

text: day

text: shown

text: in

text: red

text: which

text: clearly

text: diverges

text: from

text: the

text: expected

text: behaviour.

text: This

text: is

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [4] A. Chakraborti et al. Econophysics review: I. Empirical facts, Quanti- tative Finance, 11:7, 991-1012, 2011. [5] L. Borland, Statistical Signatures in Times of Panic: Markets as a Self- Organizing System, Quantitative Finance, 12:9, 2012. [6] L. Borland and Y. Hassid, Market panic on diﬀerent time-scales. arXiv:1010.4917, 2010. [7] R. Allez and J.-P. Bouchaud. Individual and collective stock dynamics: intra-day seasonalities, New J. Phys., 13 025010, 2011. [8] P. Gopikrishnan, M. Meyer, L.A. Amaral and H.E. Stanley, Inverse cubic law for the probability distribution of stock price variations. Eur. Phys. J. B, 3, 139-140, 1998. [9] Gopikrishnan, P., Plerou, V., Amaral, L.A., Meyer, M. and Stanley, H.E., Scaling of the distribution of ﬂuctuations of ﬁnancial market in- dices. Phys. Rev. E, 60, 5305-5316, 1999. [10] L. Kullmann, J. Toyli, J. Kertesz, A. Kanto and K. Kaski, Characteristic times in stock market indices. Physica A, 269, 98-110, 1999. [11] T. Kaizoji. A precursor of market crashes: Empirical laws of Japan’s internet bubble. Eur. Phys. J. B 50, 123-127, 2006. [12] F. Abergel et al. Financial bubbles analysis with a cross-sectional esti- mator, http://ssrn.com/abstract=1474612, 2009. [13] J.-P. Bouchaud and M. Potters. Theory of ﬁnancial risk and derivative pricing: from statistical physics to risk management. Cambridge Uni- versity Press, 2003. [14] J.-P. Bouchaud and M. Potters. Financial Applications of Random Ma- trix Theory: a short review. arXiv:0910.1205, 2009. [15] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. Amaral, T. Guhr, and H. E. Stanley. Random matrix approach to cross correlations in ﬁnancial data. Physical Review E, 65(6), 066126, 2002. [16] A. Utsugi, K. Ino, and M. Oshikawa. Random matrix theory analysis of cross correlations in ﬁnancial markets. Physical Review E, 70(2), 026110, 2004. 18

text: A.

text: Chakraborti

text: et

text: al.

text: Econophysics

text: review:

text: I.

text: Empirical

text: facts,

text: Quanti-

text: tative

text: Finance,

text: L.

text: Borland,

text: Statistical

text: Signatures

text: in

text: Times

text: of

text: Panic:

text: Markets

text: as

text: a

text: Self-

text: Organizing

text: System,

text: Quantitative

text: Finance,

text: L.

text: Borland

text: and

text: Y.

text: Hassid,

text: Market

text: panic

text: on

text: diﬀerent

text: time-scales.

text: R.

text: Allez

text: and

text: J.-P.

text: Bouchaud.

text: Individual

text: and

text: collective

text: stock

text: dynamics:

text: intra-day

text: seasonalities,

text: New

text: J.

text: Phys.,

text: P.

text: Gopikrishnan,

text: M.

text: Meyer,

text: L.A.

text: Amaral

text: and

text: H.E.

text: Stanley,

text: Inverse

text: cubic

text: law

text: for

text: the

text: probability

text: distribution

text: of

text: stock

text: price

text: variations.

text: Eur.

text: Phys.

text: J.

text: B,

text: Gopikrishnan,

text: P.,

text: Plerou,

text: V.,

text: Amaral,

text: L.A.,

text: Meyer,

text: M.

text: and

text: Stanley,

text: H.E.,

text: Scaling

text: of

text: the

text: distribution

text: of

text: ﬂuctuations

text: of

text: ﬁnancial

text: market

text: in-

text: dices.

text: Phys.

text: Rev.

text: E,

text: L.

text: Kullmann,

text: J.

text: Toyli,

text: J.

text: Kertesz,

text: A.

text: Kanto

text: and

text: K.

text: Kaski,

text: Characteristic

text: times

text: in

text: stock

text: market

text: indices.

text: Physica

text: A,

text: T.

text: Kaizoji.

text: A

text: precursor

text: of

text: market

text: crashes:

text: Empirical

text: laws

text: of

text: Japan’s

text: internet

text: bubble.

text: Eur.

text: Phys.

text: J.

text: B

text: F.

text: Abergel

text: et

text: al.

text: Financial

text: bubbles

text: analysis

text: with

text: a

text: cross-sectional

text: esti-

text: mator,

text: J.-P.

text: Bouchaud

text: and

text: M.

text: Potters.

text: Theory

text: of

text: ﬁnancial

text: risk

text: and

text: derivative

text: pricing:

text: from

text: statistical

text: physics

text: to

text: risk

text: management.

text: Cambridge

text: Uni-

text: versity

text: Press,

text: J.-P.

text: Bouchaud

text: and

text: M.

text: Potters.

text: Financial

text: Applications

text: of

text: Random

text: Ma-

text: trix

text: Theory:

text: a

text: short

text: review.

text: V.

text: Plerou,

text: P.

text: Gopikrishnan,

text: B.

text: Rosenow,

text: L.

text: A.

text: Amaral,

text: T.

text: Guhr,

text: and

text: H.

text: E.

text: Stanley.

text: Random

text: matrix

text: approach

text: to

text: cross

text: correlations

text: in

text: ﬁnancial

text: data.

text: Physical

text: Review

text: E,

text: A.

text: Utsugi,

text: K.

text: Ino,

text: and

text: M.

text: Oshikawa.

text: Random

text: matrix

text: theory

text: analysis

text: of

text: cross

text: correlations

text: in

text: ﬁnancial

text: markets.

text: Physical

text: Review

text: E,