text: As a ﬁrst step, we estimate the “unconstrained” propagator curve. Then, we estimate the resilience curve using the two parameterizations presented in Section . Estimation of the unconstrained propagator curve We ﬁrst estimate G by the linear interpolation Gˆ (cid:88) − − − l 1 (t t)g + (t t )g Gˆ(t) = g 1 (t) + i+1 i − i i+1 1 (t). l [tl,T[ t t [ti,ti+1[ i+1 i i=0 · · · · · · For t , , t ﬁxed a priori, Gˆ is linear with respect to (g , , g ). Thus, one step of Newton-Raphson’s 1 l 1 l E method (see Appendix ) determines the parameters that minimize the quadratic error (Gˆ). To approximate the long-range propagator, we choose a uniform grid t = i/l with l = 20 on the interval [0, 0.2]. On the other i hand, for a zoom on the beginning of the curve, we concentrate the t ’s near zero. i Estimation of the multi-exponential resilience curve The simultaneous estimation of multiple ρ ’s being too unstable, we choose to ﬁx four components associated i to four simple characteristic time scales (the ρ ’s are expressed in inverse hours): ρ = 6 (ten minutes), i 1 ρ = 60 (one minute), ρ = 120 (thirty seconds) and ρ = 360 (ten seconds). We then assume that the 2 · · · 3 4 vector (ρ , , ρ ) is rich enough to represent all the relevant time scales in our framework, and we focus on 1 · ·4· the weights λ , , λ associated to each scale to characterize the decay of the curve. The multi-exponential 1 4 resilience given by equation (29) becomes (cid:88) 4 − R(t) = ν + λ exp( ρ t), i i (cid:80) (cid:80) i=1 − where we re-parameterize ν = γ(1 4 λ ) > 0 and λ = γλ > 0. Reciprocally, one has γ = ν + 4 λ i=1 i i i · · ·i=1 i and λ = λ /γ. Since the ρ ’s are ﬁxed, the resilience curve R(t) is linear w.r.t. the parameter (ν, λ , , λ ) i i i 1 4 that remains to be estimated, thus Newton-Raphson’s algorithm (see Appendix ) converges with a single iteration. We then select the signiﬁcant ρ ’s as follows: i · · · 1. A ﬁrst estimation yields a “full” parameter (ν, λ , , λ ). Some of the resulting λ ’s may be non- 1 4 i positive, which is incompatible with the model. ≤ 2. While there exists i such that λ 0, we remove the ρ corresponding to the minimal λ , and we launch i i i the algorithm again with one less parameter. 3. Eventually, we have selected one to four “signiﬁcant” ρ ’s, of associated weights λ ’s that are positive, i i and the estimation process is complete. Since each of these steps only take one iteration of Newton-Raphson’s algorithm, the whole estimation protocol for the multi-exponential curve is quite fast. Therefore, in order to estimate the market adjustment lag Ladj, we can conduct the estimation several times for Ladj on some discrete grid, and compare the regression r2’s as deﬁned by (31). The result associated to the maximal r2 gives the parameters γmulti, λmulti and ρmulti for the multi-exponential resilience, along with the adjustment lag Ladj. Estimation of the mono-exponential resilience curve The multi-exponential estimation presented above serves as a starting point for the following. The market adjustment lag Ladj is already estimated, along with the associated set of parameters γmulti, λmulti, ρmulti for the multi-exponential resilience curve. We set 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

text: a

text: ﬁrst

text: step,

text: we

text: estimate

text: the

text: “unconstrained”

text: propagator

text: curve.

text: Then,

text: we

text: estimate

text: the

text: resilience

text: curve

text: using

text: the

text: two

text: parameterizations

text: presented

text: in

text: Section

text: Estimation

text: of

text: the

text: unconstrained

text: propagator

text: curve

text: We

text: ﬁrst

text: estimate

text: G

text: by

text: the

text: linear

text: interpolation

text: Gˆ

text: l

text: (t

text: t)g

text: (t

text: t

text: )g

text: Gˆ(t)

text: g

text: (t)

text: i

text: i

text: (t).

text: l

text: [tl,T[

text: t

text: t

text: i

text: For

text: t

text: t

text: ﬁxed

text: a

text: priori,

text: Gˆ

text: is

text: linear

text: with

text: respect

text: to

text: (g

text: g

text: Thus,

text: one

text: step

text: of

text: Newton-Raphson’s

text: l

text: l

text: E

text: method

text: (see

text: Appendix

text: determines

text: the

text: parameters

text: that

text: minimize

text: the

text: quadratic

text: error

text: (Gˆ).

text: To

text: approximate

text: the

text: long-range

text: propagator,

text: we

text: choose

text: a

text: uniform

text: grid

text: t

text: i/l

text: with

text: l

text: on

text: the

text: interval

text: On

text: the

text: other

text: i

text: hand,

text: for

text: a

text: zoom

text: on

text: the

text: beginning

text: of

text: the

text: curve,

text: we

text: concentrate

text: the

text: t

text: ’s

text: near

text: zero.

text: i

text: Estimation

text: of

text: the

text: multi-exponential

text: resilience

text: curve

text: The

text: simultaneous

text: estimation

text: of

text: multiple

text: ρ

text: ’s

text: being

text: too

text: unstable,

text: we

text: choose

text: to

text: ﬁx

text: four

text: components

text: associated

text: i

text: to

text: four

text: simple

text: characteristic

text: time

text: scales

text: (the

text: ρ

text: ’s

text: are

text: expressed

text: in

text: inverse

text: hours):

text: ρ

text: (ten

text: minutes),

text: i

text: ρ

text: (one

text: minute),

text: ρ

text: (thirty

text: seconds)

text: and

text: ρ

text: (ten

text: seconds).

text: We

text: then

text: assume

text: that

text: the

text: vector

text: (ρ

text: ρ

text: is

text: rich

text: enough

text: to

text: represent

text: all

text: the

text: relevant

text: time

text: scales

text: in

text: our

text: framework,

text: and

text: we

text: focus

text: on

text: ·4·

text: the

text: weights

text: λ

text: λ

text: associated

text: to

text: each

text: scale

text: to

text: characterize

text: the

text: decay

text: of

text: the

text: curve.

text: The

text: multi-exponential

text: resilience

text: given

text: by

text: equation

text: becomes

text: R(t)

text: ν

text: λ

text: exp(

text: ρ

text: t),

text: i

text: i

text: where

text: we

text: re-parameterize

text: ν

text: λ

text: and

text: λ

text: γλ

text: Reciprocally,

text: one

text: has

text: γ

text: ν

text: λ

text: i

text: i

text: i

text: i

text: and

text: λ

text: λ

text: /γ.

text: Since

text: the

text: ρ

text: ’s

text: are

text: ﬁxed,

text: the

text: resilience

text: curve

text: R(t)

text: is

text: linear

text: w.r.t.

text: the

text: parameter

text: (ν,

text: λ

text: λ

text: i

text: i

text: i

text: that

text: remains

text: to

text: be

text: estimated,

text: thus

text: Newton-Raphson’s

text: algorithm

text: (see

text: Appendix

text: converges

text: with

text: a

text: single

text: iteration.

text: We

text: then

text: select

text: the

text: signiﬁcant

text: ρ

text: ’s

text: as

text: follows:

text: i

text: A

text: ﬁrst

text: estimation

text: yields

text: a

text: “full”

text: parameter

text: (ν,

text: λ

text: λ

text: Some

text: of

text: the

text: resulting

text: λ

text: ’s

text: may

text: be

text: non-

text: i

text: positive,

text: which

text: is

text: incompatible

text: with

text: the

text: model.

text: While

text: there

text: exists

text: i

text: such

text: that

text: λ

text: we

text: remove

text: the

text: ρ

text: corresponding

text: to

text: the

text: minimal

text: λ

text: and

text: we

text: launch

text: i

text: i

text: i

text: the

text: algorithm

text: again

text: with

text: one

text: less

text: parameter.

text: Eventually,

text: we

text: have

text: selected

text: one

text: to

text: four

text: “signiﬁcant”

text: ρ

text: ’s,

text: of

text: associated

text: weights

text: λ

text: ’s

text: that

text: are

text: positive,

text: i

text: i

text: and

text: the

text: estimation

text: process

text: is

text: complete.

text: Since

text: each

text: of

text: these

text: steps

text: only

text: take

text: one

text: iteration

text: of

text: Newton-Raphson’s

text: algorithm,

text: the

text: whole

text: estimation

text: protocol

text: for

text: the

text: multi-exponential

text: curve

text: is

text: quite

text: fast.

text: Therefore,

text: in

text: order

text: to

text: estimate

text: the

text: market

text: adjustment

text: lag

text: Ladj,

text: we

text: can

text: conduct

text: the

text: estimation

text: several

text: times

text: for

text: Ladj

text: on

text: some

text: discrete

text: grid,

text: and

text: compare

text: the

text: regression

text: as

text: deﬁned

text: by

text: The

text: result

text: associated

text: to

text: the

text: maximal

text: gives

text: the

text: parameters

text: γmulti,

text: λmulti

text: and

text: ρmulti

text: for

text: the

text: multi-exponential

text: resilience,

text: along

text: with

text: the

text: adjustment

text: lag

text: Ladj.

text: Estimation

text: of

text: the

text: mono-exponential

text: resilience

text: curve

text: The

text: multi-exponential

text: estimation

text: presented

text: above

text: serves

text: as

text: a

text: starting

text: point

text: for

text: the

text: following.

text: The

text: market

text: adjustment

text: lag

text: Ladj

text: is

text: already

text: estimated,

text: along

text: with

text: the

text: associated

text: set

text: of

text: parameters

text: γmulti,

text: λmulti,

text: ρmulti

text: for

text: the

text: multi-exponential

text: resilience

text: curve.

text: We

text: set

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: 4.2 Simulated data We ﬁrst give in Tables 2 and 3 the results of our calibration protocol on two datasets simulated with the price model (4). In each table, the ﬁrst column gives the “real” simulation parameters and the second gives the estimated ones. Both datasets are composed of 150 independent realizations of the price process on two-hour windows, and we choose simulation parameters close to what is found further for stock data in order to obtain relevant benchmarks. Note that Simulation 1 features a non-zero Brownian volatility, whereas Simulation 2 is generated by the “pure” propagator model without noise. Year Simu. Calib. Year Simu. Calib. Marks type Volume Volume Ladj (sec) 4 4 βmulti 60/360 60/360 γmulti 2.70 2.35 wmulti 0.100/0.900 0.102/0.898 ρmulti 60/360 6/60/360 κ∞multi 15.0 15.2 λmulti 0.50/0.10 0.13/0.35/0.11 φsmulti 110.5/19.5 109.8/20.9 νmulti 0.40 0.41 φLcmulti 66.5−/3.5 59.7/9.7 σmulti 0.1−000 0.1917 multi − 3.1659 rm2 ulti − 9.554% βmono − 153.0 γmono − 2.38 κ∞mono − 16.6 ρmono − 68.2 φsmono − 68.7/13.1 λmono − 0.55 φLcmono − 37.4/6.1 σmono − 0.1923 mono 3.1560 rm2 ono 9.519% BR 0.833 0.839 DBR 0.250 0.257 Table 2: Calibration of the resilience (left) and intensity (right) for Simulation 1. For the φ’s, the ﬁrst entry is the constant term and the second one is the linear term. Year Simu. Calib. Year Simu. Calib. Marks type Volume Volume Ladj (sec) −2 2 βmulti 120/360 6/120/360 γmulti − 3.05 wmulti 0.050/0.950 0.0007/0.0505/0.9488 ρmulti − 6/120 κ∞multi 40.0 39.1 λmulti − 0.0005/0.6850 φsmulti 84.0/36.0 72.8/40.9 νmulti − 0.31 φLcmulti 45.0−/5.0 47.4/7.7 σmulti − 0.0055 multi − 2.7218 rm2 ulti 96.92% βmono − 82.2 γmono 3.20 3.06 κ∞mono − 19.3 ρmono 130 121.3 φsmono − 27.3/15.4 λmono 0.70 0.69 φLcmono − 17.8/2.9 σmono 0.0−000 0.0055 mono 2.6740 rm2 ono 96.92% BR 0.519 0.535 DBR 0.214 0.186 Table 3: Calibration of the resilience (left) and intensity (right) for Simulation 2. For the φ’s, the ﬁrst entry is the constant term and the second one is the linear term. Overall, the accuracy of the estimation is satisfying. The estimated Hawkes parameters are very close to the real ones, although the dimensionality is high. Importantly, the branching ratios and directional ± branching ratios are all determined accurately, within a precision of 0.03 on our experiments. Concerning the propagator, the results are more noisy for Simulation 1, which is not surprising since it includes some Brownian noise. Still, the proportion of transient impact is nearly exact and the dominant time scale is well 19 ##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## ##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## ##LTLine## ##LTLine##

label: section

text: Simulated

text: data

text: We

label: paragraph

text: ﬁrst

text: give

text: in

text: Tables

text: and

text: the

text: results

text: of

text: our

text: calibration

text: protocol

text: on

text: two

text: datasets

text: simulated

text: with

text: the

text: price

text: model

text: In

text: each

text: table,

text: the

text: ﬁrst

text: column

text: gives

text: the

text: “real”

text: simulation

text: parameters

text: and

text: the

text: second

text: gives

text: the

text: estimated

text: ones.

text: Both

text: datasets

text: are

text: composed

text: of

text: independent

text: realizations

text: of

text: the

text: price

text: process

text: on

text: two-hour

text: windows,

text: and

text: we

text: choose

text: simulation

text: parameters

text: close

text: to

text: what

text: is

text: found

text: further

text: for

text: stock

text: data

text: in

text: order

text: to

text: obtain

text: relevant

text: benchmarks.

text: Note

text: that

text: Simulation

text: features

text: a

text: non-zero

text: Brownian

text: volatility,

text: whereas

text: Simulation

text: is

text: generated

text: by

text: the

text: “pure”

text: propagator

text: model

text: without

text: noise.

text: Year

text: Simu.

text: Calib.

text: Year

text: Simu.

text: Calib.

text: Marks

text: type

text: Volume

text: Volume

text: Ladj

text: (sec)

text: βmulti

text: γmulti

text: wmulti

text: ρmulti

text: κ∞multi

text: λmulti

text: φsmulti

text: νmulti

text: φLcmulti

text: σmulti

text: multi

text: ulti

text: βmono

text: γmono

text: κ∞mono

text: ρmono

text: φsmono

text: λmono

text: φLcmono

text: σmono

text: mono

text: ono

text: BR

text: DBR

text: Table

text: Calibration

text: of

text: the

text: resilience

text: (left)

text: and

text: intensity

text: (right)

text: for

text: Simulation

text: For

text: the

text: φ’s,

text: the

text: ﬁrst

text: entry

text: is

text: the

text: constant

text: term

text: and

text: the

text: second

text: one

text: is

text: the

text: linear

text: term.

text: Year

text: Simu.

text: Calib.

text: Year

text: Simu.

text: Calib.

text: Marks

text: type

text: Volume

text: Volume

text: Ladj

text: (sec)

text: −2

text: βmulti

text: γmulti

text: wmulti

text: ρmulti

text: κ∞multi

text: λmulti

text: φsmulti

text: νmulti

text: φLcmulti

text: σmulti

text: multi

text: ulti

text: βmono

text: γmono

text: κ∞mono

text: ρmono

text: φsmono

text: λmono

text: φLcmono

text: σmono

text: mono

text: ono

text: BR

text: DBR

text: Table

text: Calibration

text: of

text: the

text: resilience

text: (left)

text: and

text: intensity

text: (right)

text: for

text: Simulation

text: For

text: the

text: φ’s,

text: the

text: ﬁrst

text: entry

text: is

text: the

text: constant

text: term

text: and

text: the

text: second

text: one

text: is

text: the

text: linear

text: term.

text: Overall,

text: the

text: accuracy

text: of

text: the

text: estimation

text: is

text: satisfying.

text: The

text: estimated

text: Hawkes

text: parameters

text: are

text: very

text: close

text: to

text: the

text: real

text: ones,

text: although

text: the

text: dimensionality

text: is

text: high.

text: Importantly,

text: the

text: branching

text: ratios

text: and

text: directional

text: branching

text: ratios

text: are

text: all

text: determined

text: accurately,

text: within

text: a

text: precision

text: of

text: on

text: our

text: experiments.

text: Concerning

text: the

text: propagator,

text: the

text: results

text: are

text: more

text: noisy

text: for

text: Simulation

text: which

text: is

text: not

text: surprising

text: since

text: it

text: includes

text: some

text: Brownian

text: noise.

text: Still,

text: the

text: proportion

text: of

text: transient

text: impact

text: is

text: nearly

text: exact

text: and

text: the

text: dominant

text: time

text: scale

text: is

text: well

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: stneve fo rebmun eht fo noitcnuf CA 02.0 01.0 50.0 20.0 10.0 stneve fo rebmun eht fo noitcnuf CA 02.0 01.0 50.0 20.0 10.0 0.2 0.5 1.0 2.0 5.0 0.2 0.5 1.0 2.0 5.0 Lag (min) Lag (min) (a) 2012 (b) 2013 Figure 3: Auto-correlation function of the number of midpoint moves triggered by trades (plain line), in log- log scale, for BNP Paribas. The (blue) dashed line is the auto-correlation generated by the mono-exponential Hawkes model, the (green) dot-dashed line is generated by the multi-exponential Hawkes model. ≈ be ruled out. Here, the standard branching ratio BR 80% is high, but the directional branching ratio DBR ≈ 40% is quite low, which is due to a non-negligible part of cross-excitation in the order ﬂow. It implies that − ≈ − ≈ − the equilibrium condition is violated since 1 ν 60% for 2012 and 1 ν 80% for 2013. Since 1 ν > DBR holds in both cases, we ﬁnd that the price process is mean-reverting on average, rather than diﬀusive. This should lead to the existence of PMS in practice, which is the object of Section . 4.4 Total Tables 7, 8 and 9 present our estimation results for the French stock Total on the periods January-September 2012 and January-September 2013. Ladj (sec) 0 2 4 6 r2 (2012) 23.093% 23.166% 23.137% 23.108% multi r2 (2013) 11.604% 11.613% 11.608% 11.606% multi Table 7: Regression r2 for the multi-exponential resilience curve, evaluated for several market adjustment lags Ladj = 0, 2, 4, 6 seconds, for the stock Total. Marks type Unit Volume Price jump L Lmono(2012) 2.2981 2.3034 2.2965 mono(2013) 2.2065 2.2127 2.2063 Table 8: Log-likelihood per point for the mono-exponential Hawkes model, evaluated for several types of marks: unit, volumes and price jumps (see eq. (34)), for the stock Total. The qualitative interpretation of the results is similar to that of Section . Yet, one should note the following points that are observable on Table 9. First, we notice that there is no signiﬁcant diﬀerence between the mono and multi-exponential propagator. Here, contrary to the BNP Paribas case, the ﬁt is slightly better ≈ ≈ with two time scales. Second, the branching ratio BR 60% and the directional branching ratio DBR 30% 23 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: stneve

font: default

text: fo

text: rebmun

text: eht

text: fo

text: noitcnuf

text: CA

text: stneve

text: fo

text: rebmun

text: eht

text: fo

text: noitcnuf

text: CA

text: Lag

text: (min)

text: Lag

text: (min)

text: (a)

text: (b)

text: Figure

text: Auto-correlation

label: caption

text: function

text: of

text: the

text: number

text: of

text: midpoint

text: moves

text: triggered

text: by

text: trades

text: (plain

text: line),

text: in

text: log-

text: log

text: scale,

text: for

text: BNP

text: Paribas.

text: The

text: (blue)

text: dashed

text: line

text: is

text: the

text: auto-correlation

text: generated

text: by

text: the

text: mono-exponential

text: Hawkes

text: model,

text: the

text: (green)

text: dot-dashed

text: line

text: is

text: generated

text: by

text: the

text: multi-exponential

text: Hawkes

text: model.

text: be

text: ruled

text: out.

text: Here,

text: the

text: standard

text: branching

text: ratio

text: BR

text: is

text: high,

text: but

text: the

text: directional

text: branching

text: ratio

text: DBR

text: is

text: quite

text: low,

text: which

text: is

text: due

text: to

text: a

text: non-negligible

text: part

text: of

text: cross-excitation

text: in

text: the

text: order

text: ﬂow.

text: It

text: implies

text: that

text: the

text: equilibrium

text: condition

text: is

text: violated

text: since

text: ν

text: for

text: and

text: ν

text: for

text: Since

text: ν

text: DBR

text: holds

text: in

text: both

text: cases,

text: we

text: ﬁnd

text: that

text: the

text: price

text: process

text: is

text: mean-reverting

text: on

text: average,

text: rather

text: than

text: diﬀusive.

text: This

text: should

text: lead

text: to

text: the

text: existence

text: of

text: PMS

text: in

text: practice,

text: which

text: is

text: the

text: object

text: of

text: Section

text: Total

text: Tables

text: and

text: present

text: our

text: estimation

text: results

text: for

text: the

text: French

text: stock

text: Total

text: on

text: the

text: periods

text: January-September

text: and

text: January-September

text: Ladj

text: (sec)

text: multi

text: multi

text: Table

text: Regression

text: for

text: the

text: multi-exponential

text: resilience

text: curve,

text: evaluated

text: for

text: several

text: market

text: adjustment

text: lags

text: Ladj

text: seconds,

text: for

text: the

text: stock

text: Total.

text: Marks

text: type

text: Unit

text: Volume

text: Price

text: jump

text: L

text: Table

text: Log-likelihood

text: per

text: point

text: for

text: the

text: mono-exponential

text: Hawkes

text: model,

text: evaluated

text: for

text: several

text: types

text: of

text: marks:

text: unit,

text: volumes

text: and

text: price

text: jumps

text: (see

text: eq.

text: for

text: the

text: stock

text: Total.

text: The

text: qualitative

text: interpretation

text: of

text: the

text: results

text: is

text: similar

text: to

text: that

text: of

text: Section

text: Yet,

text: one

text: should

text: note

text: the

text: following

text: points

text: that

text: are

text: observable

text: on

text: Table

text: First,

text: we

text: notice

text: that

text: there

text: is

text: no

text: signiﬁcant

text: diﬀerence

text: between

text: the

text: mono

text: and

text: multi-exponential

text: propagator.

text: Here,

text: contrary

text: to

text: the

text: BNP

text: Paribas

text: case,

text: the

text: ﬁt

text: is

text: slightly

text: better

text: with

text: two

text: time

text: scales.

text: Second,

text: the

text: branching

text: ratio

text: BR

text: and

text: the

text: directional

text: branching

text: ratio

text: DBR

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text: 1. To quantify repulsion in the Dyson non-colliding dynamics achieved in the diﬀusion limit in order to exhibit non-degeneracy of constraint in this regime. ∞ 2. To compute series of Plancherel measures for Sp( ) that reﬂect the intermediate representa- tion theory of the non-semisimple odd symplectic groups, and subsequently investigate their role in the correlation structure and local limit behavior of symplectic Plancherel growth. In pursuing the ﬁrst goal, we resolve a basic version of the conjecture posed after Proposition 2 of [12], conﬁrm the decompositions (4), (29) in [47] when the interlaced system starts at the origin, and thus also lift the restriction in [27] that the limiting system must start in the interior of the chamber. To show that the amount of constraint in the pre-limit system does not degenerate in this regime, we need to quantify the repulsion between particles coinciding in the limit. The path property that assumes this role is interesting in its own right: it delineates a lower envelope at t = 0 for a multivariate self-similar process with positive components that do not drift to inﬁnity R almost surely. For such -valued Markov processes drifting to inﬁnity, see [21] for rather general + techniques and integral tests. The results of Section at the heart of this development are new in this literature and are inspired by Burdzy–Kang–Ramanan [20]; we otherwise briskly follow the elegant procedures of Gorin-Shkolnikov [27] and Warren [47]. For the related case of vicious random walkers, see the series of results by Katori, Tanemura, et.al., whose paper [31] covers convergence of our even levels. For the second goal, the non-semisimplicity of odd symplectic groups is a potentially serious ∞ obstacle for the identiﬁcation of suitable “intermediate” Plancherel measures for Sp( ) – such measures can no longer arise as coeﬃcients of irreducible characters in the orthogonal decomposition of restricted inﬁnite extreme characters. Circumventing this lack of irreducibility relies on the independent works of Proctor [41] and Shtepin [43], and we suspect there are more connections in this vein to be made between probability theory and nonclassical representation theoretic objects. Defosseux [23] and Warren–Windridge [48] have already considered our dynamics for symplectic Gelfand-Tsetlin patterns, with the latter paper anticipating our interest, but the results we derive here are much in the same spirit as Borodin–Kuan [14], who cover the orthogonal case. Obtaining reﬁned local limit behavior in their manner is well codiﬁed, originating with a law of large numbers result for the limit shape of random Young tableaux, derived independently by Vershik–Kerov [46] and Logan–Shepp [37]. The former pair were studying asymptotics of the maximum dimension of irreducible representations for the symmetric group, and the latter were pursuing a variational problem arising in combinatorics posed by Richard Stanley. Baik–Deift–Johansson [3], and shortly after Okounkov [38], derived more detailed results concerning local behavior at the edge of the limit shape, inspiring much research on such behavior, including the present work; for a perfunctory list, consider [3, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 34, 35, 37, 38, 39, 40, 46]. Important Notational Convention. Quantities with indices that overﬂow are set to 0 and those ∞ with indices that underﬂow are set to . 1.2 Main applications 1.2.1 Diﬀusive scaling limit: non-degeneracy of constrained dynamics ≥ The basic description of symplectic Plancherel growth (considered up to level n 1) readily W W leads to the candidate diﬀusive limiting system ( (t)) ≥ with components k satisfying t 0 i 4

label: list

text: To

text: quantify

text: repulsion

text: in

text: the

text: Dyson

text: non-colliding

text: dynamics

text: achieved

text: in

text: the

text: diﬀusion

text: limit

text: in

text: order

text: to

text: exhibit

text: non-degeneracy

text: of

text: constraint

text: in

text: this

text: regime.

text: To

text: compute

text: series

text: of

text: Plancherel

text: measures

text: for

text: Sp(

text: that

text: reﬂect

text: the

text: intermediate

text: representa-

text: tion

text: theory

text: of

text: the

text: non-semisimple

text: odd

text: symplectic

text: groups,

text: and

text: subsequently

text: investigate

text: their

text: role

text: in

text: the

text: correlation

text: structure

text: and

text: local

text: limit

text: behavior

text: of

text: symplectic

text: Plancherel

text: growth.

text: In

text: pursuing

text: the

text: ﬁrst

text: goal,

text: we

text: resolve

text: a

text: basic

text: version

text: of

text: the

text: conjecture

text: posed

text: after

text: Proposition

text: of

text: conﬁrm

text: the

text: decompositions

text: in

text: when

text: the

text: interlaced

text: system

text: starts

text: at

text: the

text: origin,

text: and

text: thus

text: also

text: lift

text: the

text: restriction

text: in

text: that

text: the

text: limiting

text: system

text: must

text: start

text: in

text: the

text: interior

text: of

text: the

text: chamber.

text: To

text: show

text: that

text: the

text: amount

text: of

text: constraint

text: in

text: the

text: pre-limit

text: system

text: does

text: not

text: degenerate

text: in

text: this

text: regime,

text: we

text: need

text: to

text: quantify

text: the

text: repulsion

text: between

text: particles

text: coinciding

text: in

text: the

text: limit.

text: The

text: path

text: property

text: that

text: assumes

text: this

text: role

text: is

text: interesting

text: in

text: its

text: own

text: right:

text: it

text: delineates

text: a

text: lower

text: envelope

text: at

text: t

text: for

text: a

text: multivariate

text: self-similar

text: process

text: with

text: positive

text: components

text: that

text: do

text: not

text: drift

text: to

text: inﬁnity

text: R

text: almost

text: surely.

text: For

text: such

text: -valued

text: Markov

text: processes

text: drifting

text: to

text: inﬁnity,

text: see

text: for

text: rather

text: general

text: techniques

text: and

text: integral

text: tests.

text: The

text: results

text: of

text: Section

text: at

text: the

text: heart

text: of

text: this

text: development

text: are

text: new

text: in

text: this

text: literature

text: and

text: are

text: inspired

text: by

text: Burdzy–Kang–Ramanan

text: we

text: otherwise

text: briskly

text: follow

text: the

text: elegant

text: procedures

text: of

text: Gorin-Shkolnikov

text: and

text: Warren

text: For

text: the

text: related

text: case

text: of

text: vicious

text: random

text: walkers,

text: see

text: the

text: series

text: of

text: results

text: by

text: Katori,

text: Tanemura,

text: et.al.,

text: whose

text: paper

text: covers

text: convergence

text: of

text: our

text: even

text: levels.

text: For

text: the

text: second

text: goal,

text: the

text: non-semisimplicity

text: of

text: odd

text: symplectic

text: groups

text: is

text: a

text: potentially

text: serious

text: obstacle

text: for

text: the

text: identiﬁcation

text: of

text: suitable

text: “intermediate”

text: Plancherel

text: measures

text: for

text: Sp(

text: such

text: measures

text: can

text: no

text: longer

text: arise

text: as

text: coeﬃcients

text: of

text: irreducible

text: characters

text: in

text: the

text: orthogonal

text: decomposition

text: of

text: restricted

text: inﬁnite

text: extreme

text: characters.

text: Circumventing

text: this

text: lack

text: of

text: irreducibility

text: relies

text: on

text: the

text: independent

text: works

text: of

text: Proctor

text: and

text: Shtepin

text: and

text: we

text: suspect

text: there

text: are

text: more

text: connections

text: in

text: this

text: vein

text: to

text: be

text: made

text: between

text: probability

text: theory

text: and

text: nonclassical

text: representation

text: theoretic

text: objects.

text: Defosseux

text: and

text: Warren–Windridge

text: have

text: already

text: considered

text: our

text: dynamics

text: for

text: symplectic

text: Gelfand-Tsetlin

text: patterns,

text: with

text: the

text: latter

text: paper

text: anticipating

text: our

text: interest,

text: but

text: the

text: results

text: we

text: derive

text: here

text: are

text: much

text: in

text: the

text: same

text: spirit

text: as

text: Borodin–Kuan

text: who

text: cover

text: the

text: orthogonal

text: case.

text: Obtaining

text: reﬁned

text: local

text: limit

text: behavior

text: in

text: their

text: manner

text: is

text: well

text: codiﬁed,

text: originating

text: with

text: a

text: law

text: of

text: large

text: numbers

text: result

text: for

text: the

text: limit

text: shape

text: of

text: random

text: Young

text: tableaux,

text: derived

text: independently

text: by

text: Vershik–Kerov

text: and

text: Logan–Shepp

text: The

text: former

text: pair

text: were

text: studying

text: asymptotics

text: of

text: the

text: maximum

text: dimension

text: of

text: irreducible

text: representations

text: for

text: the

text: symmetric

text: group,

text: and

text: the

text: latter

text: were

text: pursuing

text: a

text: variational

text: problem

text: arising

text: in

text: combinatorics

text: posed

text: by

text: Richard

text: Stanley.

text: Baik–Deift–Johansson

text: and

text: shortly

text: after

text: Okounkov

text: derived

text: more

text: detailed

text: results

text: concerning

text: local

text: behavior

text: at

text: the

text: edge

text: of

text: the

text: limit

text: shape,

text: inspiring

text: much

text: research

text: on

text: such

text: behavior,

text: including

text: the

text: present

text: work;

text: for

text: a

text: perfunctory

text: list,

text: consider

text: Important

text: Notational

text: Convention.

text: Quantities

text: with

text: indices

text: that

text: overﬂow

text: are

text: set

text: to

text: and

text: those

text: with

text: indices

text: that

text: underﬂow

text: are

text: set

text: to

text: Main

text: applications

text: Diﬀusive

text: scaling

text: limit:

text: non-degeneracy

text: of

text: constrained

text: dynamics

text: The

text: basic

text: description

text: of

text: symplectic

text: Plancherel

text: growth

text: (considered

text: up

text: to

text: level

text: n

text: readily

text: W

text: W

text: leads

text: to

text: the

text: candidate

text: diﬀusive

text: limiting

text: system

text: (t))

text: with

text: components

text: k

text: satisfying

text: t

text: i


text: (cid:18) (cid:19) (cid:18) (cid:19) For the double integral term (65), we again use the identities (32) to compute − − z+ z 1(cid:2) v + v 1 (cid:3) J  J m (dz) s,αn  2 (cid:104) t,αm 2 α (cid:105)  − − − − − − − − − = −4d88iziiddzzzzz(cid:104)(cid:104)(cid:2)(((zzzsssvvvtt)t))√√zzv√zvv+−(z((zzs−−vstsv)vtvz)t)√√+√vzzvz+−(cid:3)szvz−−stsvzv1−v−tt√z1z√1zvvs+v+ztzsvzsvv−−vzt−t√√√√zvvzz−v(cid:105)(cid:105)11 vzv11//122/z++2−+zvzv−−−−111//122/2 iiifff αααnnn === α−αmmαm===−12 1212 4dizz (cid:104)(zsvt) zv + (z−svt)√vz − z−(cid:105)sv−t √1zv − zsv−t vz z1/v2−+vz−−11/2 if αn = −αm = −12  − − − − − − − = −4d88iziiddzzzzz(cid:104)(cid:104)2−(z22−((zszv−tss)vv√√tt)vz)vz√+zv+2−2zz−2szvs−v−stv√t−z1z1tvvz(cid:105)√√1vzvvz−11(cid:105)//zv22−v++111/zvz2−−−+11zv//−22−11/2 iiifff αααnnn === α−αmmαm===−12 1212 4dizz 2(z−svt)√vz − 2z−sv−t √1zv vz z1/v2−+vz−−11/2 if αn = −αm = −12 (68) → − where the second equality follows from the substitution z z 1 applied to the integrals deter- mined by the ﬁrst and fourth summands in each case. Now we make the additional transformation → − v v 1 applied to the integrals of the second terms in each case, which completes the v-contour ± into a positively oriented loop through the points e iθ, as indicated by the dotted line in Figure 6. (cid:18) (cid:19) (cid:18) (cid:19) Most importantly, we have a double integral over two loops with integrand − − −− 2αn+1/2 z + z 1 v + v 1 1 v 2 2π2i Js,αn 2 Jt,αm 2 z + z−1 −(v + v−1)dvmα(dz)  − − = 21πi z2−πsivzt z + z−11 −− v(v−2+ v−1)dvdz · v√√√z√vzvzvvzz−zvvzzv1111////−vz2222−−11++++vzzvzv−−−−−−111111////2222 iiiiffff ααααnnnn ==== α−α−mmααmm====−12 −1212 12 (69) The part of the integrand depending on the hydrodynamic scaling parameter N becomes − − e N(S(z0) S(z)) − , (70) eN(S(v) S(z0)) (cid:18) (cid:19) where z is to be determined and where, as in (9), 0 − − z + z 1 − z + z 1 − S(z) = S (z) := τ ν log z + η log 1 . τ,ν,η 2 2 To identify the correct steepest descent/ascent deformations, we ﬁrst rely on Proposition 5.1 of [14] to determine θ and z : 0 (cid:115) (cid:18) (cid:19) ≥ (cid:18) (cid:19) Proposition 6.2. (Proposition 5.1 of [14]) For τ/η 0, deﬁne τ − τ2 5τ ± τ2 4η 3/2 q±(τ, η) = q± := + + 1 1 + η 2η2 η 2η2 τ 38 ##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: equation

text: For

text: the

text: double

text: integral

text: term

text: we

text: again

text: use

text: the

text: identities

text: to

text: compute

text: z+

text: z

text: v

text: v

text: J

text: J

text: m

text: (dz)

text: s,αn

text: t,αm

text: α

text: iiifff

text: αααnnn

text: zv

text: (z−svt)√vz

text: zsv−t

text: vz

text: if

text: αn

text: −αm

text: iiifff

text: αααnnn

text: vz

text: if

text: αn

text: −αm

text: where

text: the

text: second

text: equality

text: follows

text: from

text: the

text: substitution

text: z

text: z

text: applied

text: to

text: the

text: integrals

text: deter-

text: mined

text: by

text: the

text: ﬁrst

text: and

text: fourth

text: summands

text: in

text: each

text: case.

text: Now

text: we

text: make

text: the

text: additional

text: transformation

text: v

text: v

text: applied

text: to

text: the

text: integrals

text: of

text: the

text: second

text: terms

text: in

text: each

text: case,

text: which

text: completes

text: the

text: v-contour

text: into

text: a

text: positively

text: oriented

text: loop

text: through

text: the

text: points

text: e

text: iθ,

text: as

text: indicated

text: by

text: the

text: dotted

text: line

text: in

text: Figure

text: Most

text: importantly,

text: we

text: have

text: a

text: double

text: integral

text: over

text: two

text: loops

text: with

text: integrand

text: z

text: z

text: v

text: v

text: v

text: Js,αn

text: Jt,αm

text: z

text: −(v

text: z

text: iiiiffff

text: ααααnnnn

text: The

text: part

text: of

text: the

text: integrand

text: depending

text: on

text: the

text: hydrodynamic

text: scaling

text: parameter

text: N

text: becomes

text: e

text: S(z))

text: eN(S(v)

text: where

text: z

text: is

text: to

text: be

text: determined

text: and

text: where,

text: as

text: in

text: z

text: z

text: z

text: z

text: S(z)

text: S

text: (z)

text: τ

text: ν

text: log

text: z

text: η

text: log

text: τ,ν,η

text: To

text: identify

text: the

text: correct

text: steepest

text: descent/ascent

text: deformations,

text: we

text: ﬁrst

text: rely

text: on

text: Proposition

text: of

text: to

text: determine

text: θ

text: and

text: z

text: Proposition

text: (Proposition

text: of

text: For

text: τ/η

text: deﬁne

text: τ

text: τ2

text: 5τ

text: τ2

text: 4η

text: q±(τ,

text: η)

text: q±

text: η

text: η

text: τ

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text: ≤ (cid:48) but set q− to 0 if 1/2 τ/η. Let z , z be the largest, smallest real roots of S (z), and let q be  (cid:48) H \(cid:8)Dmax min (cid:9) 0 the complex root ofS (z) in . Also deﬁne  (cid:8) (cid:9) D ∈ R ≥ ·  frozen :=(cid:8) (τ, ν, η) 3 : ν η q (τ, η)(cid:9) D ∈ R + · + · liquid := (τ, ν, η) 3 : η q−(τ, η) < ν < η q (τ, η) . D ∈ R+ ≤ · + empty := (τ, ν, η) 3: ν η q−(τ, η) +   Then the quantity ∈ D z , (τ, ν, η) empty min ∈ D z = z (τ, ν, η) := q , (τ, ν, η) liquid 0 0 0 ∈ D z , (τ, ν, η) frozen max | | − is critical for S(z) and satisﬁes z > 1 in the ﬁrst case; q > 1 in the second; and z < 1 in max 0 min the last. D D D Our work below will conﬁrm that the deﬁnitions of the regions liquid, frozen, and empty given here in this proposition are consistent with the introduction’s. 4 2 z contour 0 unit circle - v contour 2 - 4 - - 4 2 0 2 4 − Figure 6: The yellow region signiﬁes the region R((S(v) S(z ))) > 0 and the white elsewhere. 0 The dotted blue line indicates the completion of the v-contour into a loop by the transformation → − v v 1. ∈ D ± For (τ, ν, η) liquid, choose θ so that e iθ are points on the unit circle in the region of z where − | | the condition R((S(z ) S(z))) > 0 holds. Since z > 1, we enlarge the z-unit disk to be a steepest 0 − − 0 descent loop staying in the region R((S(z ) S(z))) < 0 but passing through the critical points 0 z and z . Similarly, make the steepest ascent v-loop pass through the points z , z and stay in the 0 0 − 0 0 region R((S(v) S(z ))) > 0. Figure 6 exempliﬁes these deformations. The double integral (65) is 0 → ∞ − now seen to vanish as N , but leaves behind residues. By inspection, the residue at v = z 1 | | → ∞ involves the term vs+t with v < 1 for Lebesgue almost all v and hence vanishes as N . The expression (69) tells us the residues at v = z are 39 ##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## ##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: but

text: set

text: q−

text: to

text: if

text: τ/η.

text: Let

text: z

text: z

text: be

text: the

text: largest,

text: smallest

text: real

text: roots

text: of

text: S

text: (z),

text: and

text: let

text: q

text: be

text: H

text: min

text: the

text: complex

text: root

text: ofS

text: (z)

text: in

text: Also

text: deﬁne

text: D

text: R

text: frozen

text: (τ,

text: ν,

text: η)

text: ν

text: η

text: q

text: (τ,

text: D

text: R

text: liquid

text: (τ,

text: ν,

text: η)

text: η

text: q−(τ,

text: η)

text: ν

text: η

text: q

text: (τ,

text: η)

text: D

text: R+

text: empty

text: (τ,

text: ν,

text: η)

text: ν

text: η

text: q−(τ,

text: η)

text: Then

text: the

text: quantity

text: D

text: z

text: (τ,

text: ν,

text: η)

text: empty

text: min

text: D

text: z

text: z

text: (τ,

text: ν,

text: η)

text: q

text: (τ,

text: ν,

text: η)

text: liquid

text: D

text: z

text: (τ,

text: ν,

text: η)

text: frozen

text: max

text: is

text: critical

text: for

text: S(z)

text: and

text: satisﬁes

text: z

text: in

text: the

text: ﬁrst

text: case;

text: q

text: in

text: the

text: second;

text: and

text: z

text: in

text: max

text: min

text: the

text: last.

text: D

text: D

text: D

text: Our

text: work

text: below

text: will

text: conﬁrm

text: that

text: the

text: deﬁnitions

text: of

text: the

text: regions

text: liquid,

text: frozen,

text: and

text: empty

text: given

text: here

text: in

text: this

text: proposition

text: are

text: consistent

text: with

text: the

text: introduction’s.

text: z

text: contour

text: unit

text: circle

text: v

text: contour

text: Figure

text: The

text: yellow

text: region

text: signiﬁes

text: the

text: region

text: R((S(v)

text: S(z

text: and

text: the

text: white

text: elsewhere.

text: The

text: dotted

text: blue

text: line

text: indicates

text: the

text: completion

text: of

text: the

text: v-contour

text: into

text: a

text: loop

text: by

text: the

text: transformation

text: v

text: v

text: D

text: For

text: (τ,

text: ν,

text: η)

text: liquid,

text: choose

text: θ

text: so

text: that

text: e

text: iθ

text: are

text: points

text: on

text: the

text: unit

text: circle

text: in

text: the

text: region

text: of

text: z

text: where

text: the

text: condition

text: R((S(z

text: S(z)))

text: holds.

text: Since

text: z

text: we

text: enlarge

text: the

text: z-unit

text: disk

text: to

text: be

text: a

text: steepest

text: descent

text: loop

text: staying

text: in

text: the

text: region

text: R((S(z

text: S(z)))

text: but

text: passing

text: through

text: the

text: critical

text: points

text: z

text: and

text: z

text: Similarly,

text: make

text: the

text: steepest

text: ascent

text: v-loop

text: pass

text: through

text: the

text: points

text: z

text: z

text: and

text: stay

text: in

text: the

text: region

text: R((S(v)

text: S(z

text: Figure

text: exempliﬁes

text: these

text: deformations.

text: The

text: double

text: integral

text: is

text: now

text: seen

text: to

text: vanish

text: as

text: N

text: but

text: leaves

text: behind

text: residues.

text: By

text: inspection,

text: the

text: residue

text: at

text: v

text: z

text: involves

text: the

text: term

text: vs+t

text: with

text: v

text: for

text: Lebesgue

text: almost

text: all

text: v

text: and

text: hence

text: vanishes

text: as

text: N

text: The

text: expression

text: tells

text: us

text: the

text: residues

text: at

text: v

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text: ecaps ecaps ecaps ecaps (cid:40) ∞ W [0, ), if k = 1 k is a Brownian motion Bk reﬂected in the interval W − W − ≥ (2) i i [ k 1, k− 1], if k 2 i i 1 ≤ ≤ ≤ ≤ for 1 k n, 1 i r , where the Bk are independent and started at 0; note our notational k i ∞ convention puts additional static particles at 0 (for index overﬂow) and at (for index underﬂow). Section makes the notion of reﬂecting Brownian movements in a time-dependent domain rigorous, W but we say the coupleddynamics of ( (t))t≥0 are non-degenerate if its components uniquely solve the system of equations  − Wk = Bik + 12L0Wik L0(W−ik−−11−Wik), i = rk with k od≤d ≤ , (3) i Bik + L0(Wk−Wk−1) L0(Wk−−1−Wk), otherwise for 1 i rk i i i 1 i ≡ where L0 is twice the semimartingale local time at 0 of a semimartingale Y and where L0∞ 0 Y (see Remark 4.2 of [20] for an explanation of the appropriate local time scalings). Intuitively, these W ≥ equations indicate the amount of constraint in the system ( (t)) ≤ ≤ is ﬁnite for every T 0. But W 0 t T demonstrating non-degeneracy of ( (t)) ≥ requires that we resolve a subtle yet concrete diﬃculty. t 0 Figure 3: The ﬁrst path in black is a standard brownian motion reﬂected at 0. Reﬂecting the blue path to stay between 0 antidmtehe black path requires an “inﬁnite amounttim” eof constraining to achieve. Figure 4: Diﬀusion limit of ﬁrst three particles nearest the wall (visualized in the single state space R ). The level 1 particle in red creates a repulsion between the wall at 0 and the level 2 particle in + time time black; although this extra room may still not be enough, we show the reﬂected level 3 wall-particle in blue has non-degenerate constrained dynamics. 5 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: ecaps

text: ecaps

text: ecaps

text: ecaps

text: W

text: if

text: k

text: k

text: is

text: a

text: Brownian

text: motion

text: Bk

text: reﬂected

text: in

text: the

text: interval

text: W

text: W

text: i

text: i

text: k

text: k−

text: if

text: k

text: i

text: i

text: for

text: k

text: n,

text: i

text: r

text: where

text: the

text: Bk

text: are

text: independent

text: and

text: started

text: at

text: note

text: our

text: notational

text: k

text: i

text: convention

text: puts

text: additional

text: static

text: particles

text: at

text: (for

text: index

text: overﬂow)

text: and

text: at

text: (for

text: index

text: underﬂow).

text: Section

text: makes

text: the

text: notion

text: of

text: reﬂecting

text: Brownian

text: movements

text: in

text: a

text: time-dependent

text: domain

text: rigorous,

text: W

text: but

text: we

text: say

text: the

text: coupleddynamics

text: of

text: are

text: non-degenerate

text: if

text: its

text: components

text: uniquely

text: solve

text: the

text: system

text: of

text: equations

text: Wk

text: Bik

text: i

text: rk

text: with

text: k

text: od≤d

text: i

text: Bik

text: otherwise

text: for

text: i

text: rk

text: i

text: i

text: i

text: i

text: where

text: is

text: twice

text: the

text: semimartingale

text: local

text: time

text: at

text: of

text: a

text: semimartingale

text: Y

text: and

text: where

text: Y

text: (see

text: Remark

text: of

text: for

text: an

text: explanation

text: of

text: the

text: appropriate

text: local

text: time

text: scalings).

text: Intuitively,

text: these

text: W

text: equations

text: indicate

text: the

text: amount

text: of

text: constraint

text: in

text: the

text: system

text: (t))

text: is

text: ﬁnite

text: for

text: every

text: T

text: But

text: W

text: t

text: T

text: demonstrating

text: non-degeneracy

text: of

text: (t))

text: requires

text: that

text: we

text: resolve

text: a

text: subtle

text: yet

text: concrete

text: diﬃculty.

text: t

text: Figure

text: The

text: ﬁrst

text: path

text: in

text: black

text: is

text: a

text: standard

text: brownian

text: motion

text: reﬂected

text: at

text: Reﬂecting

text: the

text: blue

text: path

text: to

text: stay

text: between

text: antidmtehe

text: black

text: path

text: requires

text: an

text: “inﬁnite

text: amounttim”

text: eof

text: constraining

text: to

text: achieve.

text: Figure

text: Diﬀusion

text: limit

text: of

text: ﬁrst

text: three

text: particles

text: nearest

text: the

text: wall

text: (visualized

text: in

text: the

text: single

text: state

text: space

text: R

text: The

text: level

text: particle

text: in

text: red

text: creates

text: a

text: repulsion

text: between

text: the

text: wall

text: at

text: and

text: the

text: level

text: particle

text: in

text: time

text: time

text: black;

text: although

text: this

text: extra

text: room

text: may

text: still

text: not

text: be

text: enough,

text: we

text: show

text: the

text: reﬂected

text: level

text: wall-particle

text: in

text: blue

text: has

text: non-degenerate

text: constrained

text: dynamics.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: − − of Figure 7 do not produce residues at v = z 1 since z (τ, ν, η) = z < 1 and also no residues 0 min at v = z by considering the indicated regions of the plot. Hence, the double contour vanishes as → ∞ N without leaving behind any term(cid:73)s and we are left with the triangular matrix governed by the ﬁrst term in Lemma 6.1: − − − 2rm rn( 1)αn αm − − − − 1 ≥ z(t+rm+αm) (s+rn+αn) 1(1 z)n mdz. (n m) 2πi | | z =1 Since the diagonal entries of this triangular matrix are 1, the determinant of Kγ converges to 1. D This conﬁrms the deﬁnition of frozen. ∈ D Finally, for (τ, ν, η) empty, again refrain from deforming the initial u contour, but make the − − substitutions x = (z + z 1)/2, u = (v + v 1)/2, as before. We may perform the same calculation → − leading to (68), but now the transformation v v 1 to the second summand at the end of (68) leads to a loop inside the unit circle (currently the z contour). This second summand disappears without leaving anything behind since we may shrink its v-loop arbitrarily to avoid residues at − v = z 1 while making z-deformations a(cid:73)s in Figure 7. But these deformations in the ﬁrst summand of (68) leave behind a full loop of residues at v = z similar to (71) and Lemma 6.3: − − − −2rm rn( 1)αn αm − − − − z(t+rm+αm) (s+rn+αn) 1(1 z)n mdz, 2πi | | z =1 Since we did not perform any deformation(cid:73)s to the initial u contour, we are asymptotically left with the sum of this last residue and the ﬁrst term in Lemma 6.1: − − − − 2rm rn( 1)αn αm − − − − 1 z(t+rm+αm) (s+rn+αn) 1(1 z)n mdz (n<m) 2πi | | z =1 Since the diagonal entries of this triangular matrix are 0, the determinant of Kγ converges to 0. D This conﬁrms the deﬁnition of empty. 6.2 Edge limits 6.2.1 Finite distance from the wall: Jacobi kernel ∼ · ∼ · Assume again that γ N τ > 0 and r , r N η > 0, but that s, t are ﬁxed and ﬁnite. − n m − A−ssume only the d(cid:90)iﬀe(cid:48)(cid:73)rence n m is of constant order. Let A(z) := τz + η log(1 z) and note 1 η/τ is a zero of A (z). Write our kernel as − − − − − 2αn+1/2 1 1 e (cid:10)N(A(1 η/τ) A(x)) (cid:11) (x 1)rn (1 x)αn(1 + x)1/2 π 2πi − eN(A(u)−A(1−η/τ)) Js,αn(x)Jt,αm(u)(u − 1)rm x − u dudx 1 − − + 1 ≥ J , (x 1)rn rmJ . (n m) s,αn t,αm αn Now deform the u-contour, as in Figure 8, to be a steepest ascent loop remaining in the region R − − − − − (A(u) A(1 η/τ)) > 0 and passing through the critical point 1 η/τ. If 1 η/τ < 1, the R − − unit interval already lies in the region (A(1 η/τ) A(x)) > 0, so the double integral term tends to zero without picking up residues. As for the frozen region above, the kernel converges to − − a triangular matrix whose diagona(cid:90)l entries are 1’s. But if 1 η/τ > 1, then we acquire a residue at u = x given by − − − −2αn+1/2( 1)rn rm 1 η/τ − − J (x)J (x)(1 x)rn rm+αn(1 + x)1/2dx π − s,αn t,αm 1 41 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: Figure

text: do

text: not

text: produce

text: residues

text: at

text: v

text: z

text: since

text: z

text: (τ,

text: ν,

text: η)

text: z

text: and

text: also

text: no

text: residues

text: min

text: at

text: v

text: z

text: by

text: considering

text: the

text: indicated

text: regions

text: of

text: the

text: plot.

text: Hence,

text: the

text: double

text: contour

text: vanishes

text: as

text: N

text: without

text: leaving

text: behind

text: any

text: and

text: we

text: are

text: left

text: with

text: the

text: triangular

text: matrix

text: governed

text: by

text: the

text: ﬁrst

text: term

text: in

text: Lemma

text: rn(

text: αm

text: z(t+rm+αm)

text: (s+rn+αn)

text: z)n

text: mdz.

text: (n

text: m)

text: z

text: Since

text: the

text: diagonal

text: entries

text: of

text: this

text: triangular

text: matrix

text: are

text: the

text: determinant

text: of

text: Kγ

text: converges

text: to

text: D

text: This

text: conﬁrms

text: the

text: deﬁnition

text: of

text: frozen.

text: D

text: Finally,

text: for

text: (τ,

text: ν,

text: η)

text: empty,

text: again

text: refrain

text: from

text: deforming

text: the

text: initial

text: u

text: contour,

text: but

text: make

text: the

text: substitutions

text: x

text: (z

text: z

text: u

text: (v

text: v

text: as

text: before.

text: We

text: may

text: perform

text: the

text: same

text: calculation

text: leading

text: to

text: but

text: now

text: the

text: transformation

text: v

text: v

text: to

text: the

text: second

text: summand

text: at

text: the

text: end

text: of

text: leads

text: to

text: a

text: loop

text: inside

text: the

text: unit

text: circle

text: (currently

text: the

text: z

text: contour).

text: This

text: second

text: summand

text: disappears

text: without

text: leaving

text: anything

text: behind

text: since

text: we

text: may

text: shrink

text: its

text: v-loop

text: arbitrarily

text: to

text: avoid

text: residues

text: at

text: v

text: z

text: while

text: making

text: z-deformations

text: in

text: Figure

text: But

text: these

text: deformations

text: in

text: the

text: ﬁrst

text: summand

text: of

text: leave

text: behind

text: a

text: full

text: loop

text: of

text: residues

text: at

text: v

text: z

text: similar

text: to

text: and

text: Lemma

text: rn(

text: αm

text: z(t+rm+αm)

text: (s+rn+αn)

text: z)n

text: mdz,

text: z

text: Since

text: we

text: did

text: not

text: perform

text: any

text: to

text: the

text: initial

text: u

text: contour,

text: we

text: are

text: asymptotically

text: left

text: with

text: the

text: sum

text: of

text: this

text: last

text: residue

text: and

text: the

text: ﬁrst

text: term

text: in

text: Lemma

text: rn(

text: αm

text: z(t+rm+αm)

text: (s+rn+αn)

text: z)n

text: mdz

text: (n<m)

text: z

text: Since

text: the

text: diagonal

text: entries

text: of

text: this

text: triangular

text: matrix

text: are

text: the

text: determinant

text: of

text: Kγ

text: converges

text: to

text: D

text: This

text: conﬁrms

text: the

text: deﬁnition

text: of

text: empty.

text: Edge

text: limits

text: Finite

text: distance

text: from

text: the

text: wall:

text: Jacobi

text: kernel

text: Assume

text: again

text: that

text: γ

text: N

text: τ

text: and

text: r

text: r

text: N

text: η

text: but

text: that

text: s,

text: t

text: are

text: ﬁxed

text: and

text: ﬁnite.

text: n

text: m

text: A−ssume

text: only

text: the

text: n

text: m

text: is

text: of

text: constant

text: order.

text: Let

text: A(z)

text: τz

text: η

text: z)

text: and

text: note

text: η/τ

text: is

text: a

text: zero

text: of

text: A

text: (z).

text: Write

text: our

text: kernel

text: as

text: e

text: η/τ)

text: A(x))

text: (x

text: π

text: Js,αn(x)Jt,αm(u)(u

text: x

text: u

text: dudx

text: J

text: (x

text: rmJ

text: (n

text: m)

text: s,αn

text: t,αm

text: αn

text: Now

text: deform

text: the

text: u-contour,

text: as

text: in

text: Figure

text: to

text: be

text: a

text: steepest

text: ascent

text: loop

text: remaining

text: in

text: the

text: region

text: R

text: (A(u)

text: η/τ))

text: and

text: passing

text: through

text: the

text: critical

text: point

text: η/τ.

text: If

text: η/τ

text: the

text: R

text: unit

text: interval

text: already

text: lies

text: in

text: the

text: region

text: η/τ)

text: A(x))

text: so

text: the

text: double

text: integral

text: term

text: tends

text: to

text: zero

text: without

text: picking

text: up

text: residues.

text: As

text: for

text: the

text: frozen

text: region

text: above,

text: the

text: kernel

text: converges

text: to

text: a

text: triangular

text: matrix

text: whose

text: entries

text: are

text: But

text: if

text: η/τ

text: then

text: we

text: acquire

text: a

text: residue

text: at

text: u

text: x

text: given

text: by

text: rm

text: η/τ

text: J

text: (x)J

text: x)rn

text: π

text: s,αn

text: t,αm

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − V × where q := 2k 1. Take a X(j) matrix Φj with entries denoted Φj(v , xj) =: Φj − (xj)1 ≤ , k { ∗ s |t ∈ rj }s t (s rj) whose nonzero rows determin(cid:88)e a basis for the row space (φ − φ[qs,j))(v , x) x X(j) j of (76), qs 1 s s=1 and which is biorthogonal to the Ψi in the sense ≤ ≤ Ψm − (k)Φm − (k) = δ , 1 m n. ≥ rm s rm t st k 0 × ≤ ≤ Deﬁne an r r , 1 j n, matrix Bj to have entries j j ∗ ∗ ≤ ≤ Bj := (φ − φ[qs,j) Ψj − )(v ), 1 s, t r . st qs 1 rj t s j × Add 0’s to extend Bj to an r r matrix Bj, which acts as a change of basis in the sense n n 0 ∗ ∗ ∗ (cid:48) B0jΦj = [φ0 φ[1,j), φ2 φ[3,j), . . . , φqrj−1 φ[qrj,j), 0, 0, . . .] (cid:88) ↔ ≤ ≤ (note we identify v i). Note further that for 1 t r , i j ∗ ∗ ∗ Brjj,t = (φqrj−1 φ[qrj,j) Ψjrj−t)(vrj) = (φqrj−1 Ψqrrjj−t)(vrj) = ≥ Ψqrrjj−t(x) = δrj,t x 0 where the second equality uses W Ψn = Ψi (by Proposition 3.3) and the last uses orthogonal [i,n) − decomposition and J − (1) = 1. This implies in particular that if Bj 1 is upper triangular for ≥ (cid:32) k, 1/2 (cid:33) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) j 2, then so is (cid:88)Bj. Hence, Bn is upper triangular and since Bn = Bn = M by deﬁnition, we 0 n have − j Bj ∗ −1 (Bj)−1 ∗(cid:48) Bj 0 M 1 E − W = ∗ BjΦj = ∗(cid:48) Φj = Φj. n k 1 [k,j) 0 0 0 0 0 k=1 (cid:34) (cid:32) (cid:33)(cid:35) We have thus computed th(cid:2)e (i, j)-blo(cid:3)ck (75) o(cid:88)f the kernel to be j − − − W 1 + W Ψn M 1 E − W = φ[i,j)1 + ΨiΦj. [i,j) (i<j) [i,n) k 1 [k,j) (i<j) k=1 (See [11] to ﬁnd a slightly more complicated expression for the kernel when the upper triangularity (cid:101) of M fails.) The entrywise statement reads as follows. n ∈ Z × Z X Proposition A.2. For any (s, i), (t, j) ≥ , the point process ω determined by (73) has 0 >0 (cid:88) determinantal correlations with kernel − rj K((s, i), (t, j)) = φ[i,j)(s, t)1 + Ψi − (s)Φj − (t). (i<j) ri k rj k k=1 References [1] Alexander Altland and Martin R. Zirnbauer. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B, 55:1142–1161, Jan 1997. [2] Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. 46

text: V

text: where

text: q

text: Take

text: a

text: X(j)

text: matrix

text: Φj

text: with

text: entries

text: denoted

text: Φj(v

text: xj)

text: Φj

text: k

text: s

text: |t

text: rj

text: }s

text: t

text: (s

text: rj)

text: whose

text: nonzero

text: rows

text: a

text: basis

text: for

text: the

text: row

text: space

text: (φ

text: φ[qs,j))(v

text: x)

text: x

text: X(j)

text: j

text: of

text: qs

text: s

text: and

text: which

text: is

text: biorthogonal

text: to

text: the

text: Ψi

text: in

text: the

text: sense

text: Ψm

text: (k)Φm

text: (k)

text: δ

text: m

text: n.

text: rm

text: s

text: rm

text: t

text: st

text: k

text: Deﬁne

text: an

text: r

text: r

text: j

text: n,

text: matrix

text: Bj

text: to

text: have

text: entries

text: j

text: j

text: Bj

text: (φ

text: φ[qs,j)

text: Ψj

text: )(v

text: s,

text: t

text: r

text: st

text: qs

text: rj

text: t

text: s

text: j

text: Add

text: to

text: extend

text: Bj

text: to

text: an

text: r

text: r

text: matrix

text: Bj,

text: which

text: acts

text: as

text: a

text: change

text: of

text: basis

text: in

text: the

text: sense

text: n

text: n

text: φ2

text: φ[qrj,j),

text: (note

text: we

text: identify

text: v

text: i).

text: Note

text: further

text: that

text: for

text: t

text: r

text: i

text: j

text: Brjj,t

text: φ[qrj,j)

text: Ψjrj−t)(vrj)

text: Ψqrrjj−t)(vrj)

text: Ψqrrjj−t(x)

text: δrj,t

text: x

text: where

text: the

text: second

text: equality

text: uses

text: W

text: Ψn

text: Ψi

text: (by

text: Proposition

text: and

text: the

text: last

text: uses

text: orthogonal

text: [i,n)

text: decomposition

text: and

text: J

text: This

text: implies

text: in

text: particular

text: that

text: if

text: Bj

text: is

text: upper

text: triangular

text: for

text: k,

text: j

text: then

text: so

text: is

text: Hence,

text: Bn

text: is

text: upper

text: triangular

text: and

text: since

text: Bn

text: Bn

text: M

text: by

text: deﬁnition,

text: we

text: n

text: have

text: j

text: Bj

text: −1

text: Bj

text: M

text: E

text: W

text: BjΦj

text: Φj

text: Φj.

text: n

text: k

text: [k,j)

text: We

text: have

text: thus

text: computed

text: (i,

text: the

text: kernel

text: to

text: be

text: j

text: W

text: W

text: Ψn

text: M

text: E

text: W

text: ΨiΦj.

text: [i,j)

text: (i<j)

text: [i,n)

text: k

text: [k,j)

text: (i<j)

text: (See

text: to

text: ﬁnd

text: a

text: slightly

text: more

text: complicated

text: expression

text: for

text: the

text: kernel

text: when

text: the

text: upper

text: triangularity

text: of

text: M

text: fails.)

text: The

text: entrywise

text: statement

text: reads

text: as

text: follows.

text: n

text: Z

text: Z

text: X

text: Proposition

text: For

text: any

text: (s,

text: i),

text: (t,

text: j)

text: the

text: point

text: process

text: ω

text: determined

text: by

text: has

text: determinantal

text: correlations

text: with

text: kernel

text: rj

text: K((s,

text: i),

text: (t,

text: j))

text: φ[i,j)(s,

text: Ψi

text: (s)Φj

text: (t).

text: (i<j)

text: ri

text: k

text: rj

text: k

text: References

text: Alexander

text: Altland

text: and

text: Martin

text: R.

text: Zirnbauer.

text: Nonstandard

text: symmetry

text: classes

text: in

text: mesoscopic

text: normal-superconducting

text: hybrid

text: structures.

text: Phys.

text: Rev.

text: B,

text: Jan

text: Greg

text: W.

text: Anderson,

text: Alice

text: Guionnet,

text: and

text: Ofer

text: Zeitouni.

text: An

text: introduction

text: to

text: random

text: matrices,

text: volume

text: of

text: Cambridge

text: Studies

text: in

text: Advanced

text: Mathematics.

text: Cambridge

text: University

text: Press,

text: Cambridge,


text: { ≥ ≥ } in [27] that the limiting system must start in the interior of the chamber x . . . x . Note the ≥ 1 N path property for (6) is compelling only for N 3; in the case N = 2, the single gap is proportional to a 3-dimensional Bessel process and the result is classical. 1.2.2 Determinantal correlation kernel and local limit behavior As we saw in the last section, the diﬀusion limit primarily cares inductively about interactions of three successive levels at a time. To capture the eﬀects of correlations between particles farther apart in the system, we now view symplectic Plancherel growth as determining a point conﬁguration X (γ), γ ≥ 0, in Z≥ × Z , i.e., as a random element of 2Z≥0×Z>0. A fundamental theorem about 0 >0 this point process is that it is determinantal with an explicitly computable kernel when started from the “leftmost” initial condition (see the ﬁrst image of Figure 1). To state this result and others that ≥ follow, let Jk,1/2(x), Jk,−1/2(x), k 0 den(cid:40)ote the Chebyshev polynomials of the second and third kind, respectively, (see Section ) and write 1/2, if k even α := − . k (cid:101) 1/2, if k odd (cid:101) X X X − − Consider the recoordinatization (γ) deﬁned to have components n(γ) := n(γ)+r (n k+1), ≥ ≤ ≤ k k n n 1, 1 k r (see Section for explanation and visualization). The next theorem is a n consequence of the similar statement Theorem(cid:101) 4.1 conce(cid:101)rning the general symplectic Plancherel point process. { } X ∈ TZhe×orZem 1.2. The(cid:101)kth correlation fun(cid:101)ctions ργk k≥1 of ( (γ))γ≥0 are determinantal: for z1, . . . , zk ≥ , 0 >0 P X ⊃ { } ργ(z , . . . , z ) := ( (γ) z , . . . , z ) = det[Kγ(z , z )]k , k 1 k 1 k i j i,j=1 ∈ Z × Z where the (nonsymmetric) kernel(cid:90)Kγ (cid:73)is given explicitly, for (s, n), (t, m) ≥ , by 0 >0 − 2αn+1/2 1 1 eγx (cid:90) (1 x)rn+αn(1 + x)1/2 Kγ((s, n), (t, m)) = J (x)J (u) − − dudx π 2πi − eγu s,αn t,αm (1 u)rm(x u) 1 (7) 2αn+1/2 1 − − + 1 ≥ J (x)J (x)(1 x)rn rm+αn(1 + x)1/2dx. (n m) π − s,αn t,αm 1 − Here, the u contour is a positively oriented (i.e., counterclockwise) simple loop around [ 1, 1]. Although the correlation kernel (7) of Theorem 1.2 may appear unwieldy at ﬁrst, its double integral expression allows for asymptotic expansions leading to reﬁned limiting results. Deﬁne the D region liquid of nontriv(cid:110)ial hydrodynamic limiting behavior as the collection of locat(cid:111)ions and times with a positive chance of either seeing a particle or not: (cid:101) D ∈ R × R × R liquid := (τ, ν, η) : 0 < lim ρ[Nτ](([Nν], [Nη])) < 1 . (8) + + + →∞ 1 N Under this hDydrodynamic scaling, the method of st∈eeRpest de(cid:18)scent applied (cid:19)directly to the kernel (7) shows that liquid is exactly the region of (τ, ν, η) 3 where the function + − − z + z 1 − z + z 1 − S(z) = S (z) := τ ν log z + η log 1 . (9) τ,ν,η 2 2 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: in

text: that

text: the

text: limiting

text: system

text: must

text: start

text: in

text: the

text: interior

text: of

text: the

text: chamber

text: x

text: x

text: Note

text: the

text: N

text: path

text: property

text: for

text: is

text: compelling

text: only

text: for

text: N

text: in

text: the

text: case

text: N

text: the

text: single

text: gap

text: is

text: proportional

text: to

text: a

text: Bessel

text: process

text: and

text: the

text: result

text: is

text: classical.

text: Determinantal

text: correlation

text: kernel

text: and

text: local

text: limit

text: behavior

text: As

text: we

text: saw

text: in

text: the

text: last

text: section,

text: the

text: diﬀusion

text: limit

text: primarily

text: cares

text: inductively

text: about

text: interactions

text: of

text: three

text: successive

text: levels

text: at

text: a

text: time.

text: To

text: capture

text: the

text: eﬀects

text: of

text: correlations

text: between

text: particles

text: farther

text: apart

text: in

text: the

text: system,

text: we

text: now

text: view

text: symplectic

text: Plancherel

text: growth

text: as

text: determining

text: a

text: point

text: conﬁguration

text: X

text: (γ),

text: γ

text: in

text: Z≥

text: Z

text: i.e.,

text: as

text: a

text: random

text: element

text: of

text: A

text: fundamental

text: theorem

text: about

text: this

text: point

text: process

text: is

text: that

text: it

text: is

text: determinantal

text: with

text: an

text: explicitly

text: computable

text: kernel

text: when

text: started

text: from

text: the

text: “leftmost”

text: initial

text: condition

text: (see

text: the

text: ﬁrst

text: image

text: of

text: Figure

text: To

text: state

text: this

text: result

text: and

text: others

text: that

text: follow,

text: let

text: k

text: the

text: Chebyshev

text: polynomials

text: of

text: the

text: second

text: and

text: third

text: kind,

text: respectively,

text: (see

text: Section

text: and

text: write

text: if

text: k

text: even

text: α

text: k

text: if

text: k

text: odd

text: X

text: X

text: X

text: Consider

text: the

text: recoordinatization

text: (γ)

text: deﬁned

text: to

text: have

text: components

text: n(γ)

text: n(γ)+r

text: (n

text: k

text: k

text: n

text: n

text: k

text: r

text: (see

text: Section

text: for

text: explanation

text: and

text: visualization).

text: The

text: next

text: theorem

text: is

text: a

text: n

text: consequence

text: of

text: the

text: similar

text: statement

text: the

text: general

text: symplectic

text: Plancherel

text: point

text: process.

text: X

text: TZhe×orZem

text: correlation

text: ργk

text: of

text: are

text: determinantal:

text: for

text: zk

text: P

text: X

text: ργ(z

text: z

text: (γ)

text: z

text: z

text: det[Kγ(z

text: z

text: )]k

text: k

text: k

text: k

text: i

text: j

text: Z

text: Z

text: where

text: the

text: (nonsymmetric)

text: given

text: explicitly,

text: for

text: (s,

text: n),

text: (t,

text: m)

text: by

text: eγx

text: Kγ((s,

text: n),

text: (t,

text: m))

text: J

text: (x)J

text: (u)

text: dudx

text: π

text: eγu

text: s,αn

text: t,αm

text: u)rm(x

text: u)

text: J

text: (x)J

text: x)rn

text: (n

text: m)

text: π

text: s,αn

text: t,αm

text: Here,

text: the

text: u

text: contour

text: is

text: a

text: positively

text: oriented

text: (i.e.,

text: counterclockwise)

text: simple

text: loop

text: around

text: Although

text: the

text: correlation

text: kernel

text: of

text: Theorem

text: may

text: appear

text: unwieldy

text: at

text: ﬁrst,

text: its

text: double

text: integral

text: expression

text: allows

text: for

text: asymptotic

text: expansions

text: leading

text: to

text: reﬁned

text: limiting

text: results.

text: Deﬁne

text: the

text: D

text: region

text: liquid

text: of

text: hydrodynamic

text: limiting

text: behavior

text: as

text: the

text: collection

text: of

text: and

text: times

text: with

text: a

text: positive

text: chance

text: of

text: either

text: seeing

text: a

text: particle

text: or

text: not:

text: D

text: R

text: R

text: R

text: liquid

text: (τ,

text: ν,

text: η)

text: lim

text: ρ[Nτ](([Nν],

text: [Nη]))

text: N

text: Under

text: this

text: hDydrodynamic

text: scaling,

text: the

text: method

text: of

text: st∈eeRpest

text: applied

text: to

text: the

text: kernel

text: shows

text: that

text: liquid

text: is

text: exactly

text: the

text: region

text: of

text: (τ,

text: ν,

text: η)

text: where

text: the

text: function

text: z

text: z

text: z

text: z

text: S(z)

text: S

text: (z)

text: τ

text: ν

text: log

text: z

text: η

text: log

text: τ,ν,η

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 Diﬀusive scaling limit and repulsion of Dyson gaps 2.1 Time-dependent Skorokhod problem and Skorokhod reﬂection map The machinery of the extended Skorokhod map provides a remarkably natural and careful con- struction of our particle system and its diﬀusion limit. A solution to the Skorokhod problem (SP) of ≤ reﬂecting a particle ψ in a (potentially time-dependent) interval [(cid:96), r], (cid:96) r, is a pair (φ, η) where η ∈ “pushes” the particle ψ to remain in [(cid:96), r] and the resulting reﬂected movements φ = ψ + η [(cid:96), r] should be regular; classica(cid:90)lly, η should be of boun(cid:90)ded variation and (φ, η) should satisfy the familiar conditions ∞ ∞ 1 dη (s) = 1 dη (s) = 0, (11) ((cid:96)(s)<φ(s)) (cid:96) (φ(s)<r(s)) r 0 0 − where η , η are nondecreasing functions aﬀording the (minimal) decomposition η = η η . But (cid:96) r (cid:96) r as explained in Section , the conditions (11) are too restrictive for our situation. To state a weaker R R notion of solution, let D( , S) be the space of right-continuous functions with left limits on that + R R − R −∞+∞ take values in a separable complete metric space S. Write D := D( , ), D := D( , [ , )), R −∞ ∞ + + and D+ := D( , ( , ]). + ∈ −× × ≤ ∈ × Deﬁnition 2.1. (Deﬁnition 2.2, [20]) Given ((cid:96), r, ψ) D D+ D with (cid:96) r, a pair (φ, η) D D solves the extended Skorokhod problem (ESP) on [(cid:96), r] for ψ if ∈ ≥ 1. φ(t) = ψ(t) + η(t) [(cid:96)(t), r(t)] for t 0 ≤ ∞ 2. For 0 s < t < , − ≥ − − ≥ η(t) η(s) 0, if φ < r on (s, t]; and η(s) η(s ) 0, if φ(s) < r(s) − ≤ − − ≤ η(t) η(s) 0, if (cid:96) < φ on (s, t]; and η(s) η(s ) 0, if (cid:96)(s) < φ(s) ∈ − × × ≤ Given ((cid:96), r, ψ) D D+ D with (cid:96) r, an explicit candidate for the reﬂected process φ is (cid:18) | − (cid:19) provided by the extended Skorokhod map Γ((cid:96), r ψ) := ψ Ξ (ψ), where (cid:96),r − ∧ − − ∧ − Ξ (ψ)(t) := max (ψ(0) r(0))+ inf (ψ(u) (cid:96)(u)), sup [(ψ(s) r(s)) inf (ψ(u) (cid:96)(u))] . (cid:96),r ∈ ∈ ∈ u [0,t] s [0,t] u [s,t] ≡ ∞ − − ≥ Note some simpliﬁcations: if r , then Ξ ∞(ψ)(t) = sup ∈ [(cid:96)(s) ψ(s)]+, t 0; also, at the (cid:96), s [0,t] initial time 0, − ∧ − − Ξ (ψ)(0) = [ψ(0) r(0)]+ [ψ(0) (cid:96)(0)] . (12) (cid:96),r | In words then, Γ((cid:96), r ψ) starts by putting the particle ψ at the nearest point in [(cid:96), r] and maintains its increments except for putting those that land outside of [(cid:96), r] again at the nearest point inside. − × × Theorem 2.6 and Proposition 2.8 of [20] show Γ is a jointly continuous function on D D+ D (each topologized by uniform convergence on compacts) and provides the unique solution | | − (φ, η) := (Γ((cid:96), r ψ), Γ((cid:96), r ψ) ψ) − ≤ to the associated ESP on [(cid:96), r] for ψ. The condition inf ∈ (r(t) (cid:96)(t)) > 0, 0 t < t , is t [t1,t2] 1 2 suﬃcient for (φ, η) to satisfy the more regular conditions (11) on [t , t ]; cf. Corollary 2.4 of [20]. 1 2 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Diﬀusive

text: scaling

text: limit

text: and

text: repulsion

text: of

text: Dyson

text: gaps

text: Time-dependent

text: Skorokhod

text: problem

text: and

text: Skorokhod

text: reﬂection

text: map

text: The

text: machinery

text: of

text: the

text: extended

text: Skorokhod

text: map

text: provides

text: a

text: remarkably

text: natural

text: and

text: careful

text: con-

text: struction

text: of

text: our

text: particle

text: system

text: and

text: its

text: diﬀusion

text: limit.

text: A

text: solution

text: to

text: the

text: Skorokhod

text: problem

text: (SP)

text: of

text: reﬂecting

text: a

text: particle

text: ψ

text: in

text: a

text: (potentially

text: time-dependent)

text: interval

text: r],

text: r,

text: is

text: a

text: pair

text: (φ,

text: η)

text: where

text: η

text: “pushes”

text: the

text: particle

text: ψ

text: to

text: remain

text: in

text: r]

text: and

text: the

text: resulting

text: reﬂected

text: movements

text: φ

text: ψ

text: η

text: r]

text: should

text: be

text: regular;

text: η

text: should

text: be

text: of

text: variation

text: and

text: (φ,

text: η)

text: should

text: satisfy

text: the

text: familiar

text: conditions

text: dη

text: (s)

text: dη

text: (s)

text: (φ(s)<r(s))

text: r

text: where

text: η

text: η

text: are

text: nondecreasing

text: functions

text: aﬀording

text: the

text: (minimal)

text: decomposition

text: η

text: η

text: η

text: But

text: r

text: r

text: as

text: explained

text: in

text: Section

text: the

text: conditions

text: are

text: too

text: restrictive

text: for

text: our

text: situation.

text: To

text: state

text: a

text: weaker

text: R

text: R

text: notion

text: of

text: solution,

text: let

text: D(

text: S)

text: be

text: the

text: space

text: of

text: right-continuous

text: functions

text: with

text: left

text: limits

text: on

text: that

text: R

text: R

text: R

text: take

text: values

text: in

text: a

text: separable

text: complete

text: metric

text: space

text: S.

text: Write

text: D

text: D(

text: D

text: D(

text: R

text: and

text: D+

text: D(

text: Deﬁnition

text: (Deﬁnition

text: Given

text: r,

text: ψ)

text: D

text: D+

text: D

text: with

text: r,

text: a

text: pair

text: (φ,

text: η)

text: D

text: D

text: solves

text: the

text: extended

text: Skorokhod

text: problem

text: (ESP)

text: on

text: r]

text: for

text: ψ

text: if

text: φ(t)

text: ψ(t)

text: η(t)

text: r(t)]

text: for

text: t

text: For

text: s

text: t

text: η(t)

text: η(s)

text: if

text: φ

text: r

text: on

text: (s,

text: t];

text: and

text: η(s)

text: η(s

text: if

text: φ(s)

text: r(s)

text: η(t)

text: η(s)

text: if

text: φ

text: on

text: (s,

text: t];

text: and

text: η(s)

text: η(s

text: if

text: φ(s)

text: Given

text: r,

text: ψ)

text: D

text: D+

text: D

text: with

text: r,

text: an

text: explicit

text: candidate

text: for

text: the

text: reﬂected

text: process

text: φ

text: is

text: provided

text: by

text: the

text: extended

text: Skorokhod

text: map

text: r

text: ψ)

text: ψ

text: Ξ

text: (ψ),

text: where

text: Ξ

text: (ψ)(t)

text: max

text: inf

text: (ψ(u)

text: sup

text: [(ψ(s)

text: r(s))

text: inf

text: (ψ(u)

text: u

text: s

text: u

text: [s,t]

text: Note

text: some

text: simpliﬁcations:

text: if

text: r

text: then

text: Ξ

text: ∞(ψ)(t)

text: sup

text: ψ(s)]+,

text: t

text: also,

text: at

text: the

text: s

text: initial

text: time

text: Ξ

text: In

text: words

text: then,

text: r

text: ψ)

text: starts

text: by

text: putting

text: the

text: particle

text: ψ

text: at

text: the

text: nearest

text: point

text: in

text: r]

text: and

text: maintains

text: its

text: increments

text: except

text: for

text: putting

text: those

text: that

text: land

text: outside

text: of

text: r]

text: again

text: at

text: the

text: nearest

text: point

text: inside.

text: Theorem

text: and

text: Proposition

text: of

text: show

text: Γ

text: is

text: a

text: jointly

text: continuous

text: function

text: on

text: D

text: D+

text: D

text: (each

text: topologized

text: by

text: uniform

text: convergence

text: on

text: compacts)

text: and

text: provides

text: the

text: unique

text: solution

text: (φ,

text: η)

text: r

text: ψ),

text: r

text: ψ)

text: ψ)

text: to

text: the

text: associated

text: ESP

text: on

text: r]

text: for

text: ψ.

text: The

text: condition

text: inf

text: (r(t)

text: t

text: t

text: is

text: t

text: suﬃcient

text: for

text: (φ,

text: η)

text: to

text: satisfy

text: the

text: more

text: regular

text: conditions

text: on

text: [t

text: t

text: cf.

text: Corollary

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##