text: 5 shift current generation and provides a practical guide- An alternative expression for the shift vector, which line to search for materials with large shift currents. avoids the use of φb , can be obtained from Eq. (A1). − nm VII. ACKNOWLEDGMENTS Since σ2(0; ω, ω) is real we hav(cid:2)e (cid:3) We thank F. de Juan, M. Kolodrubetz and S. Barraza- | | − Ra,b rb 2 = Im rb rb . (A6) Lopez for useful discussions. B.M.F. acknowledges sup- nm nm mn nm;a port from AFOSR MURI, Conacyt, and NERSC Con- tract No. DE-AC02-05CH11231. T.M. acknowledges The right-hand-side is gauge invariant and has simple support from the Gordon and Betty Moore Foundation’s analytical expressions for eﬀective models of monochalco- EPiQS Initiative Theory Center Grant. J.E.M. acknowl- genides22,23. It contains two important physical eﬀects, edges funding from NSF DMR-1507141 and a Simons density of states and the geometry of Bloch wave func- Investigatorship. tions. To disentangle these eﬀects, let us consider the Appendix A: Phase-independent expression of shift (cid:54) (cid:54) case where rb = 0 (equivalently vb = 0) then the vector nm nm shift vector itself is well deﬁned, (cid:2) (cid:3) The shift current tensor, Eq. (2) in the main text, in d dimension is usually writ(cid:90)ten(cid:88)as7 Ra,b = −| 1 | Im rb rb , (A7) nm rb 2 mn nm;a nm − iπe3 σabc(0; ω, ω) = 2(cid:126)2 fnm(rmb nrncm;a and independent of the density of states. In the BZ nm independent-particle approximation, the imaginary part − + rmc nrnbm(cid:82);a)δ(ω(cid:82)mn ω), (A1) of the dielectric functio(cid:90)n, (cid:88) ≡ εab(ω) − e2π − theHeBrreilwloeuidne(cid:126)ﬁznoende t(hBeZ(cid:126)i)ntien−grd(cid:126)aldaims enBZsions dfokr/(n2oπt)adtioovnearl 2ε0 = δab ε0(cid:126) BZ nm fnmrnamrmb nδ(ωmn ω). convenience. ω = ω ω are band energy diﬀer- (A8) nm − n m ences and f = f f diﬀerences of Fermi distribution nm n m functions of band n and m. The dipole matrix elements is dominated by the second term and comparing with ra and generalized derivatives are nm Eq. (A4) we obtain Eq. (2) in the main text. Appendix B: The optical gauge ≡ (cid:54) rb Ab [n = m and 0 otherwise] (A2) nm nm The solutions of the Schrodinger equation with a peri- ≡ ∂rb − − odic potential are Bloch wavefunctions, rb nm i(Aa Aa )rb . (A3) nm;a ∂ka nn mm nm · (cid:104) | | (cid:105) (cid:126) ψ (k, r) = eik ru (k, r), (B1) Ab = i u ∂ u are the Berry connections, ω = n n (cid:126) n−m(cid:126) n ∂kb m −nm ω ω are the band energies and f = f f n m nm n m are the fermionic occupation numbers. We can write where n is the bands index and k the crystal momen- Ab = v /iω , for nondegenerate bands where nm nm nm tum. u (k, r + R) = u (k, r) is the cell periodic part vb is the velocity matrix element. Setting b = c n n nm of the wave function and R is a lattice vector. The so- for linear polarization and using polar representation, | | − lutions of the Schrodinger equation are invariant under rnam = rnam e iφanm, Eq. (A(cid:90)1) r(cid:88)educes to phase transformations [U(1) gauge transformations], − −πe3 | | (cid:48) σabb(0; ω, ω) = (cid:126)2 fnmRna,mb rnbm 2 ψn(k, r) = eiϕn(k)ψn(k, r). (B2) BZ nm × − δ(ω ω), (A4) mn Under gauge transformations the Berry connections transform as where Ra,b is the so-called shift ‘vector,’ nm (cid:48) − A b = Ab ei(ϕm ϕn) (B3) nm nm ∂φb − (cid:48) − ∂ϕ (k) Ra,b = nm + Aa Aa , (A5) A b = Ab m . (B4) nm ∂ka nn mm mm mm ∂kb ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: shift

text: current

text: generation

text: and

text: provides

text: a

text: practical

text: guide-

text: An

text: alternative

text: expression

text: for

text: the

text: shift

text: vector,

text: which

text: line

text: to

text: search

text: for

text: materials

text: with

text: large

text: shift

text: currents.

text: avoids

text: the

text: use

text: of

text: φb

text: can

text: be

text: obtained

text: from

text: Eq.

text: nm

text: VII.

text: ACKNOWLEDGMENTS

text: Since

text: ω,

text: ω)

text: is

text: real

text: we

label: equation

text: We

text: thank

text: F.

text: de

text: Juan,

text: M.

text: Kolodrubetz

text: and

text: S.

text: Barraza-

text: Ra,b

text: rb

text: Im

text: rb

text: rb

text: Lopez

text: for

text: useful

text: discussions.

text: B.M.F.

text: acknowledges

text: sup-

text: nm

text: nm

text: mn

text: nm;a

text: port

text: from

text: AFOSR

text: MURI,

text: Conacyt,

text: and

text: NERSC

text: Con-

text: tract

text: No.

text: T.M.

text: acknowledges

text: The

text: right-hand-side

text: is

text: gauge

text: invariant

text: and

text: has

text: simple

text: support

text: from

text: the

text: Gordon

text: and

text: Betty

text: Moore

text: Foundation’s

text: analytical

text: expressions

text: for

text: eﬀective

text: models

text: of

text: monochalco-

text: EPiQS

text: Initiative

text: Theory

text: Center

text: Grant.

text: J.E.M.

text: acknowl-

text: It

text: contains

text: two

text: important

text: physical

text: eﬀects,

text: edges

text: funding

text: from

text: NSF

text: and

text: a

text: Simons

text: density

text: of

text: states

text: and

text: the

text: geometry

text: of

text: Bloch

text: wave

text: func-

text: Investigatorship.

text: tions.

text: To

text: disentangle

text: these

text: eﬀects,

text: let

text: us

text: consider

text: the

text: Appendix

text: A:

text: Phase-independent

text: expression

text: of

text: shift

text: case

text: where

text: rb

text: (equivalently

text: vb

text: then

text: the

text: vector

text: nm

text: nm

text: shift

text: vector

text: itself

text: is

text: well

text: deﬁned,

text: The

text: shift

text: current

text: tensor,

text: Eq.

text: in

text: the

text: main

text: text,

text: in

text: d

text: dimension

text: is

text: usually

text: Ra,b

text: Im

text: rb

text: rb

text: nm

text: rb

text: mn

text: nm;a

text: nm

text: ω,

text: ω)

text: fnm(rmb

text: nrncm;a

text: and

text: independent

text: of

text: the

text: density

text: of

text: states.

text: In

text: the

text: BZ

text: nm

text: independent-particle

text: approximation,

text: the

text: imaginary

text: part

text: rmc

text: ω),

text: of

text: the

text: dielectric

text: εab(ω)

text: enBZsions

text: δab

text: BZ

text: nm

text: fnmrnamrmb

text: nδ(ωmn

text: ω).

text: convenience.

text: ω

text: ω

text: ω

text: are

text: band

text: energy

text: diﬀer-

text: nm

text: n

text: m

text: ences

text: and

text: f

text: f

text: f

text: diﬀerences

text: of

text: Fermi

text: distribution

text: nm

text: n

text: m

text: functions

text: of

text: band

text: n

text: and

text: m.

text: The

text: dipole

text: matrix

text: elements

text: is

text: dominated

text: by

text: the

text: second

text: term

text: and

text: comparing

text: with

text: ra

text: and

text: generalized

text: derivatives

text: are

text: nm

text: Eq.

text: we

text: obtain

text: Eq.

text: in

text: the

text: main

text: text.

text: Appendix

text: B:

text: The

text: optical

text: gauge

text: rb

text: Ab

text: [n

text: m

text: and

text: otherwise]

text: nm

text: nm

text: The

text: solutions

text: of

text: the

text: Schrodinger

text: equation

text: with

text: a

text: peri-

text: ∂rb

text: odic

text: potential

text: are

text: Bloch

text: wavefunctions,

text: rb

text: nm

text: i(Aa

text: Aa

text: )rb

text: nm;a

text: ∂ka

text: nn

text: mm

text: nm

text: ψ

text: (k,

text: r)

text: eik

text: ru

text: (k,

text: r),

text: Ab

text: i

text: u

text: u

text: are

text: the

text: Berry

text: connections,

text: ω

text: n

text: n

text: n

text: ∂kb

text: m

text: −nm

text: ω

text: ω

text: are

text: the

text: band

text: energies

text: and

text: f

text: f

text: f

text: n

text: m

text: nm

text: n

text: m

text: are

text: the

text: fermionic

text: occupation

text: numbers.

text: We

text: can

text: write

text: where

text: n

text: is

text: the

text: bands

text: index

text: and

text: k

text: the

text: crystal

text: momen-

text: Ab

text: v

text: /iω

text: for

text: nondegenerate

text: bands

text: where

text: nm

text: nm

text: nm

text: tum.

text: u

text: (k,

text: r

text: R)

text: u

text: (k,

text: r)

text: is

text: the

text: cell

text: periodic

text: part

text: vb

text: is

text: the

text: velocity

text: matrix

text: element.

text: Setting

text: b

text: c

text: n

text: n

text: nm

text: of

text: the

text: wave

text: function

text: and

text: R

text: is

text: a

text: lattice

text: vector.

text: The

text: so-

text: for

text: linear

text: polarization

text: and

text: using

text: polar

text: representation,

text: lutions

text: of

text: the

text: Schrodinger

text: equation

text: are

text: invariant

text: under

text: rnam

text: rnam

text: e

text: iφanm,

text: Eq.

text: to

text: phase

text: transformations

text: gauge

text: transformations],

text: −πe3

text: ω,

text: ω)

text: fnmRna,mb

text: rnbm

text: ψn(k,

text: r)

text: eiϕn(k)ψn(k,

text: r).

text: BZ

text: nm

text: δ(ω

text: ω),

text: mn

text: Under

text: gauge

text: transformations

text: the

text: Berry

text: connections

text: transform

text: as

text: where

text: Ra,b

text: is

text: the

text: so-called

text: shift

text: ‘vector,’

text: nm

text: A

text: b

text: Ab

text: ei(ϕm

text: ϕn)

text: nm

text: nm

text: ∂φb

text: ∂ϕ

text: (k)

text: Ra,b

text: nm

text: Aa

text: Aa

text: A

text: b

text: Ab

text: m

text: nm

text: ∂ka

text: nn

text: mm

text: mm

text: mm

text: ∂kb

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 0.8 0.8 (a) (b) 0.6 0.6 0.4 0.4 0.2 0.2 1.25 1.5 1.75 2 2.25 2.5 1.25 1.5 1.75 2 2.25 2.5 Photon energy (eV) Photon energy (eV) FIG. 4. (Color online) Transmission as a function of the incident photon energy. Panel (a) shows simulations for the BB model. Panel (b) corresponds to simulations performed using the BC model. In both panels transmission for the bare array of slits is indicated as a black line. Red lines correspond to isotropic initial conditions, green lines are data for molecules pre-aligned along x with the degree of alignment n = 3. Blue lines are results of simulations for molecules pre-aligned along y, with the same degree of alignment n = 3. The period of the array in all simulations is × − 330 nm. Other parameters for all molecular models are: a molecular concentration of 5 1026 m 3, a non-radiative decay time of 1 ps and a pure dephasing time is 600 fs. IV. SUMMARY AND CONCLUSIONS In summary we extended the conventional model of exciton-plasmon materials based on simple interacting two-level emitters by explicitly including the ro-vibrational structure of diatomic molecules using time-dependent wave packet propagation on electronic potential noissimsnarT noissimsnarT ##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##

label: paragraph

font: default

text: (a)

text: (b)

text: Photon

text: energy

text: (eV)

text: Photon

text: energy

text: (eV)

text: FIG.

text: (Color

label: caption

text: online)

text: Transmission

text: as

text: a

text: function

text: of

text: the

text: incident

text: photon

text: energy.

text: Panel

text: (a)

text: shows

text: simulations

text: for

text: the

text: BB

text: model.

text: Panel

text: (b)

text: corresponds

text: to

text: simulations

text: performed

text: using

text: the

text: BC

text: model.

text: In

text: both

text: panels

text: transmission

text: for

text: the

text: bare

text: array

text: of

text: slits

text: is

text: indicated

text: as

text: a

text: black

text: line.

text: Red

text: lines

text: correspond

text: to

text: isotropic

text: initial

text: conditions,

text: green

text: lines

text: are

text: data

text: for

text: molecules

text: pre-aligned

text: along

text: x

text: with

text: the

text: degree

text: of

text: alignment

text: n

text: Blue

text: lines

text: are

text: results

text: of

text: simulations

text: for

text: molecules

text: pre-aligned

text: along

text: y,

text: with

text: the

text: same

text: degree

text: of

text: alignment

text: n

text: The

text: period

text: of

text: the

text: array

text: in

text: all

text: simulations

text: is

text: nm.

text: Other

text: parameters

text: for

text: all

text: molecular

text: models

text: are:

text: a

text: molecular

text: concentration

text: of

text: m

text: a

text: non-radiative

text: decay

text: time

text: of

text: ps

text: and

text: a

text: pure

text: dephasing

text: time

text: is

text: fs.

text: IV.

text: SUMMARY

text: AND

text: CONCLUSIONS

text: In

text: summary

text: we

text: extended

text: the

text: conventional

text: model

text: of

text: exciton-plasmon

text: materials

text: based

text: on

text: simple

text: interacting

text: two-level

text: emitters

text: by

text: explicitly

text: including

text: the

text: ro-vibrational

text: structure

text: of

text: diatomic

text: molecules

text: using

text: time-dependent

text: wave

text: packet

text: propagation

text: on

text: electronic

text: potential

text: noissimsnarT

text: noissimsnarT

text: ##LTFigure##

text: ##LTLine##

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text: 4 where Φ (r, R) and Φ (r, R) denote the electronic wave functions associated with the g e ground and excited electronic states. The electron coordinates are denoted by r. R denotes the internuclear coordinate and we adopt Hund’s case (b) representation for the description of the electronic wave functions in the molecular frame16. The ro-vibrational time-dependent wave functions χ (R, t) and χ (R, t) are expanded on a limited set of normalized Wigner g e rotation matrices as17 (cid:88) χ (R, t) = (cid:88) χ(g) (R, t) DN ∗ (Rˆ), (3a) g N,M M,0 N,M ∗ χ (R, t) = χ(e) (R, t) DN (Rˆ) , (3b) e N,M M,0 N,M where N denotes the molecular rotational quantum number while M denotes its pro- jection on a ﬁxed axis of the laboratory frame. Introducing these expansions in the time- dependent Schro¨dinger equation describing the molecule-ﬁeld interaction and projecting onto the electronic and rotational basis functions yields, in the dipole approximation, a set of coupled diﬀerential equations14 for the nuclear wave packets χ(g) (R, t) and χ(e) (R, t) N,M N,M that we solve using the short-time split-operator method18. To integrate the correspond- ing diﬀerential equations we need to deﬁne the potential energy curves V (R) and V (R) g e associated with the ground and excited electronic states of the molecule and the ma- trix elements which couple the nuclear wave packets evolving on these electronic potential curves. Such matrix elements are given in Ref. [14] and for the potential curves we use − − − − Morse potentials with V (R) = T + D(1 exp[ a(R R )])2 D for a bound state and e − − V (R) = T +2D exp[ a(R R )] for a dissociative state. The parameters used in our numer- e ical simulations in the case of bound-bound transitions and in the case of bound-continuum transitions are given in Table I. These parameters have been chosen such that the absorption spectra of the molecules lie in the energy domain of the SPP resonance. We assume that the molecules are prepared initially in the ground vibrational level of the ground electronic state. In terms of rotation, the molecules are assumed to be initially isotropically distributed (with N = M = 0) or pre-aligned (see Section ). The essence of our non-Hermitian Schro¨dinger approach lies in the introduction of time-dependent gain and decay rates14,15 for the ground and excited electronic states. These rates can be viewed as

text: where

text: Φ

text: (r,

text: R)

text: and

text: Φ

text: (r,

text: R)

text: denote

text: the

text: electronic

text: wave

text: functions

text: associated

text: with

text: the

text: g

text: e

text: ground

text: and

text: excited

text: electronic

text: states.

text: The

text: electron

text: coordinates

text: are

text: denoted

text: by

text: r.

text: R

text: denotes

text: the

text: internuclear

text: coordinate

text: and

text: we

text: adopt

text: Hund’s

text: case

text: (b)

text: representation

text: for

text: the

text: description

text: of

text: the

text: electronic

text: wave

text: functions

text: in

text: the

text: molecular

text: The

text: ro-vibrational

text: time-dependent

text: wave

text: functions

text: χ

text: (R,

text: t)

text: and

text: χ

text: (R,

text: t)

text: are

text: expanded

text: on

text: a

text: limited

text: set

text: of

text: normalized

text: Wigner

text: g

text: e

text: rotation

text: matrices

text: χ

text: (R,

text: t)

text: χ(g)

text: (R,

text: t)

text: DN

text: (Rˆ),

text: g

text: N,M

text: N,M

text: χ

text: (R,

text: t)

text: χ(e)

text: (R,

text: t)

text: DN

text: (Rˆ)

text: e

text: N,M

text: N,M

text: where

text: N

text: denotes

text: the

text: molecular

text: rotational

text: quantum

text: number

text: while

text: M

text: denotes

text: its

text: pro-

text: jection

text: on

text: a

text: ﬁxed

text: axis

text: of

text: the

text: laboratory

text: frame.

text: Introducing

text: these

text: expansions

text: in

text: the

text: time-

text: dependent

text: Schro¨dinger

text: equation

text: describing

text: the

text: molecule-ﬁeld

text: interaction

text: and

text: projecting

text: onto

text: the

text: electronic

text: and

text: rotational

text: basis

text: functions

text: yields,

text: in

text: the

text: dipole

text: approximation,

text: a

text: set

text: of

text: coupled

text: diﬀerential

text: for

text: the

text: nuclear

text: wave

text: packets

text: χ(g)

text: (R,

text: t)

text: and

text: χ(e)

text: (R,

text: t)

text: N,M

text: N,M

text: that

text: we

text: solve

text: using

text: the

text: short-time

text: split-operator

text: To

text: integrate

text: the

text: correspond-

text: ing

text: diﬀerential

text: equations

text: we

text: need

text: to

text: deﬁne

text: the

text: potential

text: energy

text: curves

text: V

text: (R)

text: and

text: V

text: (R)

text: g

text: e

text: associated

text: with

text: the

text: ground

text: and

text: excited

text: electronic

text: states

text: of

text: the

text: molecule

text: and

text: the

text: ma-

text: trix

text: elements

text: which

text: couple

text: the

text: nuclear

text: wave

text: packets

text: evolving

text: on

text: these

text: electronic

text: potential

text: curves.

text: Such

text: matrix

text: elements

text: are

text: given

text: in

text: Ref.

text: and

text: for

text: the

text: potential

text: curves

text: we

text: use

text: Morse

text: potentials

text: with

text: V

text: (R)

text: T

text: exp[

text: a(R

text: R

text: D

text: for

text: a

text: bound

text: state

text: and

text: e

text: V

text: (R)

text: T

text: exp[

text: a(R

text: R

text: for

text: a

text: dissociative

text: state.

text: The

text: parameters

text: used

text: in

text: our

text: numer-

text: e

text: ical

text: simulations

text: in

text: the

text: case

text: of

text: bound-bound

text: transitions

text: and

text: in

text: the

text: case

text: of

text: bound-continuum

text: transitions

text: are

text: given

text: in

text: Table

text: I.

text: These

text: parameters

text: have

text: been

text: chosen

text: such

text: that

text: the

text: absorption

text: spectra

text: of

text: the

text: molecules

text: lie

text: in

text: the

text: energy

text: domain

text: of

text: the

text: SPP

text: resonance.

text: We

text: assume

text: that

text: the

text: molecules

text: are

text: prepared

text: initially

text: in

text: the

text: ground

text: vibrational

text: level

text: of

text: the

text: ground

text: electronic

text: state.

text: In

text: terms

text: of

text: rotation,

text: the

text: molecules

text: are

text: assumed

text: to

text: be

text: initially

text: isotropically

text: distributed

text: (with

text: N

text: M

text: or

text: pre-aligned

text: (see

text: Section

text: The

text: essence

text: of

text: our

text: non-Hermitian

text: Schro¨dinger

text: approach

text: lies

text: in

text: the

text: introduction

text: of

text: time-dependent

text: gain

text: and

text: decay

text: for

text: the

text: ground

text: and

text: excited

text: electronic

text: states.

text: These

text: rates

text: can

text: be

text: viewed

text: as


text: 6 where Γ is the damping parameter, ω is the bulk plasma frequency, and ε is the high- p r frequency limit of the dielectric constant. For the range of frequencies considered in this work the following set of parameters was chosen to represent silver: ε = 8.926, ω = 11.585 eV, r p and Γ = 0.203 eV19. The corresponding current density in the metal region is evaluated according to the time-dependent equation20 (cid:126) ∂J + ΓJ(cid:126) = ε ω2E(cid:126) . (5) ∂t 0 p The system is excited by a plane wave at normal incidence propagating in the negative y-direction. We employ the short pulse method10 and evaluate refection and transmission in the far-ﬁeld zone as functions of the incident photon energy. The thickness of the metal is ﬁxed at 200 nm and the slits’ width is set to 60 nm. The slits’ period is varied between 200 nm and 500 nm. It is seen in Fig. 1b that a bare slit array exhibits a sharp resonance near 1.864 eV for the period of 330 nm. The maximum of transmission at this energy indicates that this peak is a SPP resonance21. The steady-state EM intensity distribution at this frequency is shown in Fig. 1c in logarithmic scale. The ﬁeld is highly localized near the sharp edges of the slits extending both into the slits and into outer surfaces on the input and output sides. It should be noted that the transmission reaches almost 80% at the SPP resonance with a relatively high Q-factor of 4 for an SPP mode. The high Q-factor and the fact that the local ﬁeld enhancement is quite signiﬁcant promises a noticeable EM coupling to molecules placed in close vicinity of the slits. Our choice of particular parameters describing the diatomic molecules is somewhat arbi- trary as our major goal in this paper is to investigate the coupling between molecules and SPP waves on a qualitative level. Both the high sensitivity of SPP modes to various geo- metrical parameters3 and outstanding progress in nanofabrication techniques22 would allow one to construct a plasmonic system capable of coupling to a molecule of one’s choice. We consider two frequently used models for a molecule depicted in Fig. 1d with a bound ground state and either bound or dissociative excited electronic state. We refer to the ﬁrst model of bound-bound transitions as the BB-model and to the second model with bound-continuum transitions as the BC-model. In our simulations the ground state is described by a Morse potential with a minimum located near the internuclear distance R = 5.06 a.u. The bound excited state has a minimum at R = 5.89 a.u. Both bound and dissociative excited states ##LTLine##

text: where

text: Γ

text: is

text: the

text: damping

text: parameter,

text: ω

text: is

text: the

text: bulk

text: plasma

text: frequency,

text: and

text: ε

text: is

text: the

text: high-

text: p

text: r

text: frequency

text: limit

text: of

text: the

text: dielectric

text: constant.

text: For

text: the

text: range

text: of

text: frequencies

text: considered

text: in

text: this

text: work

text: the

text: following

text: set

text: of

text: parameters

text: was

text: chosen

text: to

text: represent

text: silver:

text: ε

text: ω

text: eV,

text: r

text: p

text: and

text: Γ

text: The

text: corresponding

text: current

text: density

text: in

text: the

text: metal

text: region

text: is

text: evaluated

text: according

text: to

text: the

text: time-dependent

text: ∂J

text: ε

text: ∂t

text: p

text: The

text: system

text: is

text: excited

text: by

text: a

text: plane

text: wave

text: at

text: normal

text: incidence

text: propagating

text: in

text: the

text: negative

text: y-direction.

text: We

text: employ

text: the

text: short

text: pulse

text: and

text: evaluate

text: refection

text: and

text: transmission

text: in

text: the

text: far-ﬁeld

text: zone

text: as

text: functions

text: of

text: the

text: incident

text: photon

text: energy.

text: The

text: thickness

text: of

text: the

text: metal

text: is

text: ﬁxed

text: at

text: nm

text: and

text: the

text: slits’

text: width

text: is

text: set

text: to

text: nm.

text: The

text: slits’

text: period

text: is

text: varied

text: between

text: nm

text: and

text: nm.

text: It

text: is

text: seen

text: in

text: Fig.

text: that

text: a

text: bare

text: slit

text: array

text: exhibits

text: a

text: sharp

text: resonance

text: near

text: eV

text: for

text: the

text: period

text: of

text: nm.

text: The

text: maximum

text: of

text: transmission

text: at

text: this

text: energy

text: indicates

text: that

text: this

text: peak

text: is

text: a

text: SPP

text: The

text: steady-state

text: EM

text: intensity

text: distribution

text: at

text: this

text: frequency

text: is

text: shown

text: in

text: Fig.

text: in

text: logarithmic

text: scale.

text: The

text: ﬁeld

text: is

text: highly

text: localized

text: near

text: the

text: sharp

text: edges

text: of

text: the

text: slits

text: extending

text: both

text: into

text: the

text: slits

text: and

text: into

text: outer

text: surfaces

text: on

text: the

text: input

text: and

text: output

text: sides.

text: It

text: should

text: be

text: noted

text: that

text: the

text: transmission

text: reaches

text: almost

text: at

text: the

text: SPP

text: resonance

text: with

text: a

text: relatively

text: high

text: Q-factor

text: of

text: for

text: an

text: SPP

text: mode.

text: The

text: high

text: Q-factor

text: and

text: the

text: fact

text: that

text: the

text: local

text: ﬁeld

text: enhancement

text: is

text: quite

text: signiﬁcant

text: promises

text: a

text: noticeable

text: EM

text: coupling

text: to

text: molecules

text: placed

text: in

text: close

text: vicinity

text: of

text: the

text: slits.

text: Our

text: choice

text: of

text: particular

text: parameters

text: describing

text: the

text: diatomic

text: molecules

text: is

text: somewhat

text: arbi-

text: trary

text: as

text: our

text: major

text: goal

text: in

text: this

text: paper

text: is

text: to

text: investigate

text: the

text: coupling

text: between

text: molecules

text: and

text: SPP

text: waves

text: on

text: a

text: qualitative

text: level.

text: Both

text: the

text: high

text: sensitivity

text: of

text: SPP

text: modes

text: to

text: various

text: geo-

text: metrical

text: and

text: outstanding

text: progress

text: in

text: nanofabrication

text: would

text: allow

text: one

text: to

text: construct

text: a

text: plasmonic

text: system

text: capable

text: of

text: coupling

text: to

text: a

text: molecule

text: of

text: one’s

text: choice.

text: We

text: consider

text: two

text: frequently

text: used

text: models

text: for

text: a

text: molecule

text: depicted

text: in

text: Fig.

text: with

text: a

text: bound

text: ground

text: state

text: and

text: either

text: bound

text: or

text: dissociative

text: excited

text: electronic

text: state.

text: We

text: refer

text: to

text: the

text: ﬁrst

text: model

text: of

text: bound-bound

text: transitions

text: as

text: the

text: BB-model

text: and

text: to

text: the

text: second

text: model

text: with

text: bound-continuum

text: transitions

text: as

text: the

text: BC-model.

text: In

text: our

text: simulations

text: the

text: ground

text: state

text: is

text: described

text: by

text: a

text: Morse

text: potential

text: with

text: a

text: minimum

text: located

text: near

text: the

text: internuclear

text: distance

text: R

text: a.u.

text: The

text: bound

text: excited

text: state

text: has

text: a

text: minimum

text: at

text: R

text: a.u.

text: Both

text: bound

text: and

text: dissociative

text: excited

text: states

text: ##LTLine##


text: Convergence of Siegel-Veech constants ∗ Benjamin Dozier Abstract R We show that for any weakly convergent sequence of ergodic SL ( )-invariant probability 2 measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech con- stants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-Mirzakhani-Mohammadi, this yields the (previously con- jectured) convergence of sequences of Siegel-Veech constants associated to Teichmu¨ller curves in genus two. The proof uses a recurrence result closely related to techniques developed by Eskin-Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface. 1 Introduction 1.1 Setting Basic Deﬁnitions. A translation surface is a pair X = (M, ω), where M is a Riemann surface, and ω is a holomorphic 1-form. Away from its zeroes, ω deﬁnes a ﬂat (Euclidean) metric. The metric has a conical singularity of cone angle 2(n + 1)π at each zero of order n. A saddle connection is a geodesic segment that starts and ends at zeroes (we allow the endpoints to coincide), with no zeroes on the interior of the segment. We can also consider closed loops not hitting zeroes that are geodesic with respect to the ﬂat metric. Whenever there is one of these, there will always be a continuous family of parallel closed geodesic loops with the same length. We refer to a maximal such family as a cylinder. Every cylinder is bounded by a union of saddle connections parallel to the cylinder. M M The bundle Ω of holomorphic 1-forms over (the moduli space of genus g Riemann sur- g g faces), with zero section removed, can be thought of as the moduli space of translation surfaces. This bundle breaks up into strata of translation surfaces that have the same multiplicities of the H zeroes of ω. We denote by (m , . . . , m ) the stratum of unit-area surfaces with k zeroes of order 1 k m , . . . , m . 1 k R H There is an action of SL ( ) on each stratum which will play a central role in our discussion. 2 To see the action, we ﬁrst observe that by cutting along saddle connections, we can represent every translation surface as a set of polygons in the plane, such that every side is paired up with a parallel R side of equal length. Since SL ( ) acts on polygons in the plane, preserving the property of a pair 2 H of sides being parallel and equal length, the group acts on . We will work mostly with elements of (cid:18) (cid:19) (cid:18) (cid:19) the following form: − − e t 0 cos θ sin θ g = , r = . t 0 et θ sin θ cos θ ∗ Department of Mathematics, Stanford University, bdozier@stanford.edu. Supported in part by NSF grant DGE- 114747. 1 ##LTLine##

text: Convergence

label: title

text: of

text: Siegel-Veech

text: constants

label: author

text: Benjamin

text: Dozier

text: Abstract

text: R

label: abstract

text: We

text: show

text: that

text: for

text: any

text: weakly

text: convergent

text: sequence

text: of

text: ergodic

text: SL

text: )-invariant

text: probability

text: measures

text: on

text: a

text: stratum

text: of

text: unit-area

text: translation

text: surfaces,

text: the

text: corresponding

text: Siegel-Veech

text: con-

text: stants

text: converge

text: to

text: the

text: Siegel-Veech

text: constant

text: of

text: the

text: limit

text: measure.

text: Together

text: with

text: a

text: measure

text: equidistribution

text: result

text: due

text: to

text: Eskin-Mirzakhani-Mohammadi,

text: this

text: yields

text: the

text: (previously

text: con-

text: jectured)

text: convergence

text: of

text: sequences

text: of

text: Siegel-Veech

text: constants

text: associated

text: to

text: Teichmu¨ller

text: curves

text: in

text: genus

text: two.

text: The

text: proof

text: uses

text: a

text: recurrence

text: result

text: closely

text: related

text: to

text: techniques

text: developed

text: by

text: Eskin-Masur.

text: We

text: also

text: use

text: this

text: recurrence

text: result

text: to

text: get

text: an

text: asymptotic

text: quadratic

text: upper

text: bound,

text: with

text: a

text: uniform

text: constant

text: depending

text: only

text: on

text: the

text: stratum,

text: for

text: the

text: number

text: of

text: saddle

text: connections

text: of

text: length

text: at

text: most

text: R

text: on

text: a

text: unit-area

text: translation

text: surface.

text: Introduction

text: Setting

text: Basic

text: Deﬁnitions.

text: A

text: translation

text: surface

text: is

text: a

text: pair

text: X

text: (M,

text: ω),

text: where

text: M

text: is

text: a

text: Riemann

text: surface,

text: and

text: ω

text: is

text: a

text: holomorphic

text: Away

text: from

text: its

text: zeroes,

text: ω

text: deﬁnes

text: a

text: ﬂat

text: (Euclidean)

text: metric.

text: The

text: metric

text: has

text: a

text: conical

text: singularity

text: of

text: cone

text: angle

text: at

text: each

text: zero

text: of

text: order

text: n.

text: A

text: saddle

text: connection

text: is

text: a

text: geodesic

text: segment

text: that

text: starts

text: and

text: ends

text: at

text: zeroes

text: (we

text: allow

text: the

text: endpoints

text: to

text: coincide),

text: with

text: no

text: zeroes

text: on

text: the

text: interior

text: of

text: the

text: segment.

text: We

text: can

text: also

text: consider

text: closed

text: loops

text: not

text: hitting

text: zeroes

text: that

text: are

text: geodesic

text: with

text: respect

text: to

text: the

text: ﬂat

text: metric.

text: Whenever

text: there

text: is

text: one

text: of

text: these,

text: there

text: will

text: always

text: be

text: a

text: continuous

text: family

text: of

text: parallel

text: closed

text: geodesic

text: loops

text: with

text: the

text: same

text: length.

text: We

text: refer

text: to

text: a

text: maximal

text: such

text: family

text: as

text: a

text: cylinder.

text: Every

text: cylinder

text: is

text: bounded

text: by

text: a

text: union

text: of

text: saddle

text: connections

text: parallel

text: to

text: the

text: cylinder.

text: M

text: M

text: The

text: bundle

text: Ω

text: of

text: holomorphic

text: over

text: (the

text: moduli

text: space

text: of

text: genus

text: g

text: Riemann

text: sur-

text: g

text: g

text: faces),

text: with

text: zero

text: section

text: removed,

text: can

text: be

text: thought

text: of

text: as

text: the

text: moduli

text: space

text: of

text: translation

text: surfaces.

text: This

text: bundle

text: breaks

text: up

text: into

text: strata

text: of

text: translation

text: surfaces

text: that

text: have

text: the

text: same

text: multiplicities

text: of

text: the

text: H

text: zeroes

text: of

text: ω.

text: We

text: denote

text: by

text: (m

text: m

text: the

text: stratum

text: of

text: unit-area

text: surfaces

text: with

text: k

text: zeroes

text: of

text: order

text: k

text: m

text: m

text: k

text: R

text: H

text: There

text: is

text: an

text: action

text: of

text: SL

text: on

text: each

text: stratum

text: which

text: will

text: play

text: a

text: central

text: role

text: in

text: our

text: discussion.

text: To

text: see

text: the

text: action,

text: we

text: ﬁrst

text: observe

text: that

text: by

text: cutting

text: along

text: saddle

text: connections,

text: we

text: can

text: represent

text: every

text: translation

text: surface

text: as

text: a

text: set

text: of

text: polygons

text: in

text: the

text: plane,

text: such

text: that

text: every

text: side

text: is

text: paired

text: up

text: with

text: a

text: parallel

text: R

text: side

text: of

text: equal

text: length.

text: Since

text: SL

text: acts

text: on

text: polygons

text: in

text: the

text: plane,

text: preserving

text: the

text: property

text: of

text: a

text: pair

text: H

text: of

text: sides

text: being

text: parallel

text: and

text: equal

text: length,

text: the

text: group

text: acts

text: on

text: We

text: will

text: work

text: mostly

text: with

text: elements

text: of

text: the

text: following

text: form:

text: e

text: t

text: cos

text: θ

text: sin

text: θ

text: g

text: r

text: t

text: et

text: θ

text: sin

text: θ

text: cos

text: θ

text: Department

text: of

text: Mathematics,

text: Stanford

text: University,

text: bdozier@stanford.edu.

text: Supported

text: in

text: part

text: by

text: NSF

text: grant

text: DGE-

text: ##LTLine##


text: − least n 1, it must have dimension at least n (this uses the fact that aﬃne invariant submanifolds N ∈ N come from proper immersions). So x = c( ) S ( ). This completes the proof of the claim. ⊂ ⊂ R n ≤ Now for any closed sets A B , it follows immediately from our deﬁnition that rank(A) rank(B). Then, from the deﬁnition of rank, this fact, the claim above, and the inductive assumption, N ≤ N ∗ ≤ N ≤ − − − rank(S − ( )) rank(S − ( ) ) + 1 rank(S ( )) + 1 (d n + 1) + 1 = d (n 1) + 1, n 1 n 1 n (cid:4) which completes the induction. 3 Proof of Theorem 1.1 (Convergence of Siegel-Veech con- stants) Proof of Theorem 1.1. The proof would be immediate if strata were compact, but they are not. The recurrence-type result Proposition 1.1 allows us to get around this. (cid:82) The Siegel-Veech constant can be deﬁned by (cid:82) H fˆdµ c(µ) = , R fdλ 2 for any f : R2 → R continuous and compactly s(cid:88)upported. Here fˆ : H → R is the Siegel-Veech transform of f given by fˆ(X) := f(c), ∈ c Λ(X) ⊂ R where Λ(X) 2 is the multi-set of holonomies of cylinders. The holonomy of a saddle connection C R s is the element of (which we identify with 2) given by integrating the 1-form ω along any of the periodic geodesics deﬁning the cylinder. (cid:90) (cid:90) Hence to prove Theorem 1.1, it suﬃces to show that lim fˆdµ = fˆdη n→∞ H n H for all such f. Note that if fˆ were compactly supported, this would follow immediately from the deﬁnition of weak-* convergence. The idea is to approximate fˆ by compactly supported functions, and bound the integral of the error term using integrability results from [EM01]. The key point is that we need a bound for the error term that is independent of the particular measure µ. { ∈ H ≤ } Let C = X : 1 K . These sets are compact. Now let χ be a continuous function H → K (cid:96)(X) (cid:90) H\ (cid:90) · K [0, 1] whose value is 1 on C and 0 on C . Deﬁne fˆ = fˆ χ . Note that fˆ is compactly K K+1 K K K supported, hence (cid:12) (cid:12) lim fˆ dµ = fˆ dη.(cid:12)(cid:82) (cid:82) (cid:12) n→∞ H K n H K (cid:12) (cid:12) − It remains to show that by choosing K large, we can make H fˆdµ H fˆ d(cid:82)µ uniformly small K for any choice of ergodic SL (R)-invariant probability measure µ. We can assume f, and hence fˆ, 2 are non-negative, since we only need to show the equality for some f for which H fˆdη is non-zero. Since f is compactly supported, it is dominated by some multiple of an indicator function of a large ball. By [EM01] Theorem 5.1(a), it follows that fˆ < C/(cid:96)1+δ for some C and 0 < δ < 1/2 (the cited result is about counts of saddle connections, but since every cylinder is bounded by saddle connections, we get the same bound for cylinders). Then 8 ##LTLine## ##LTLine##

text: least

text: n

text: it

text: must

text: have

text: dimension

text: at

text: least

text: n

text: (this

text: uses

text: the

text: fact

text: that

text: aﬃne

text: invariant

text: submanifolds

text: N

text: N

text: come

text: from

text: proper

text: immersions).

text: So

text: x

text: c(

text: S

text: This

text: completes

text: the

text: proof

text: of

text: the

text: claim.

text: R

text: n

text: Now

text: for

text: any

text: closed

text: sets

text: A

text: B

text: it

text: follows

text: immediately

text: from

text: our

text: deﬁnition

text: that

text: rank(A)

text: rank(B).

text: Then,

text: from

text: the

text: deﬁnition

text: of

text: rank,

text: this

text: fact,

text: the

text: claim

text: above,

text: and

text: the

text: inductive

text: assumption,

text: N

text: N

text: N

text: rank(S

text: rank(S

text: rank(S

text: (d

text: n

text: d

text: (n

text: n

text: n

text: n

text: which

text: completes

text: the

text: induction.

text: Proof

text: of

text: Theorem

text: (Convergence

text: of

text: Siegel-Veech

text: con-

text: stants)

text: Proof

text: of

text: Theorem

text: The

text: proof

text: would

text: be

text: immediate

text: if

text: strata

text: were

text: compact,

text: but

text: they

text: are

text: not.

text: The

text: recurrence-type

text: result

text: Proposition

text: allows

text: us

text: to

text: get

text: around

text: this.

text: The

text: Siegel-Veech

text: constant

text: can

text: be

text: deﬁned

text: by

text: H

text: fˆdµ

text: c(µ)

text: R

text: fdλ

text: for

text: any

text: f

text: R

text: continuous

text: and

text: compactly

text: Here

text: fˆ

text: H

text: R

text: is

text: the

text: Siegel-Veech

text: transform

text: of

text: f

text: given

text: by

text: fˆ(X)

text: f(c),

text: c

text: Λ(X)

text: R

text: where

text: Λ(X)

text: is

text: the

text: multi-set

text: of

text: holonomies

text: of

text: cylinders.

text: The

text: holonomy

text: of

text: a

text: saddle

text: connection

text: C

text: R

text: s

text: is

text: the

text: element

text: of

text: (which

text: we

text: identify

text: with

text: given

text: by

text: integrating

text: the

text: ω

text: along

text: any

text: of

text: the

text: periodic

text: geodesics

text: deﬁning

text: the

text: cylinder.

text: Hence

text: to

text: prove

text: Theorem

text: it

text: suﬃces

text: to

text: show

text: that

text: lim

text: fˆdµ

text: fˆdη

text: n→∞

text: H

text: n

text: H

text: for

text: all

text: such

text: f.

text: Note

text: that

text: if

text: fˆ

text: were

text: compactly

text: supported,

text: this

text: would

text: follow

text: immediately

text: from

text: the

text: deﬁnition

text: of

text: weak-*

text: convergence.

text: The

text: idea

text: is

text: to

text: approximate

text: fˆ

text: by

text: compactly

text: supported

text: functions,

text: and

text: bound

text: the

text: integral

text: of

text: the

text: error

text: term

text: using

text: integrability

text: results

text: from

text: The

text: key

text: point

text: is

text: that

text: we

text: need

text: a

text: bound

text: for

text: the

text: error

text: term

text: that

text: is

text: independent

text: of

text: the

text: particular

text: measure

text: µ.

text: H

text: Let

text: C

text: X

text: K

text: These

text: sets

text: are

text: compact.

text: Now

text: let

text: χ

text: be

text: a

text: continuous

text: function

text: H

text: K

text: H\

text: K

text: whose

text: value

text: is

text: on

text: C

text: and

text: on

text: C

text: Deﬁne

text: fˆ

text: fˆ

text: χ

text: Note

text: that

text: fˆ

text: is

text: compactly

text: K

text: K

text: K

text: K

text: supported,

text: hence

text: lim

text: fˆ

text: dµ

text: fˆ

text: n→∞

text: H

text: K

text: n

text: H

text: K

text: It

text: remains

text: to

text: show

text: that

text: by

text: choosing

text: K

text: large,

text: we

text: can

text: make

text: H

text: fˆdµ

text: H

text: fˆ

text: uniformly

text: small

text: K

text: for

text: any

text: choice

text: of

text: ergodic

text: SL

text: (R)-invariant

text: probability

text: measure

text: µ.

text: We

text: can

text: assume

text: f,

text: and

text: hence

text: fˆ,

text: are

text: non-negative,

text: since

text: we

text: only

text: need

text: to

text: show

text: the

text: equality

text: for

text: some

text: f

text: for

text: which

text: H

text: fˆdη

text: is

text: non-zero.

text: Since

text: f

text: is

text: compactly

text: supported,

text: it

text: is

text: dominated

text: by

text: some

text: multiple

text: of

text: an

text: indicator

text: function

text: of

text: a

text: large

text: ball.

text: By

text: Theorem

text: it

text: follows

text: that

text: fˆ

text: for

text: some

text: C

text: and

text: δ

text: (the

text: cited

text: result

text: is

text: about

text: counts

text: of

text: saddle

text: connections,

text: but

text: since

text: every

text: cylinder

text: is

text: bounded

text: by

text: saddle

text: connections,

text: we

text: get

text: the

text: same

text: bound

text: for

text: cylinders).

text: Then

text: ##LTLine##

text: ##LTLine##


text: (cid:12)(cid:90) (cid:90) (cid:12) (cid:90) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − ≤ (cid:90) ≤ 1 fˆdµ fˆ dµ fˆdµ C dµ (1) H H K H\ H\ (cid:96)1+δ CK CK ≤ 1 1 C (cid:48) (cid:48) dµ, (2) Kδ H\ (cid:96)1+δ+δ CK (cid:48) (cid:48) (cid:48) where we choose δ > 0 such that δ + δ < 1/2. We introduce this extra δ to get the 1 (cid:48) term → ∞ Kδ in the above, which will allow us to get decay as K . By [EM01] Lemma 5.5, the last integral above is ﬁnite; we need to show the somewhat stronger statement that it is bounded from above independent of the choice of µ. The idea is to replace the integral of 1 (cid:48) over the whole stratum by the integral over large (cid:96)1+δ+δ circles centered at a µ-generic point, using Nevo’s theorem, and then use Proposition 1.1 to bound the integrals over circles. To apply Nevo’s theorem, we need to choose a smoothing function φ : R → R that is non-negative, smooth, and compactly supported (having the smoothing is ﬁne for our purposes; Nevo’s result should also be true without having to smooth). (cid:90) (cid:90) (cid:90) (cid:18) (cid:90) (cid:19) ∈ H Now by Nevo’s theorem (see [EM01] Theorem 1.5) and Proposition 1.1, for µ a.e. X , ∞ ∞ 1 · (cid:90) − 1 2π 1 (cid:48) dµ φ(t)dt = l→im∞ φ(τ t) (cid:48) dθ dt H (cid:96)1+δ+δ −∞ τ ∞ −∞ 2π 0 (cid:96)(gtrθX)1+δ+δ ≤ (cid:82) b φ(t)dt. −∞ ≤ Hence H 1 (cid:48) dµ b, for any µ(cid:12).(cid:90)Plugging(cid:90)into (2) (cid:12)gives, for all µ, (cid:96)1+δ+δ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − ≤ · b H fˆdµ H fˆKdµ C Kδ(cid:48) . (3) · (cid:48) Now we put everything together. Fix (cid:15) > 0. Choose K large so that C b/Kδ < (cid:15)/3. Now choose N such that (cid:12)(cid:90) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − fˆ dµ fˆ dη < (cid:15)/3 (4) H K n H K ≥ (cid:12)(cid:90)for all n (cid:90)N. (cid:12) (cid:12)(cid:90) (cid:90) (cid:12) (cid:12)(cid:90) ≥ (cid:90) (cid:12) (cid:12)(cid:90) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)Then by the triang(cid:12)le i(cid:12)nequality, and (3), (4)(cid:12), for(cid:12) n N, (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − ≤ − − − fˆdµ fˆdη fˆdµ fˆ dµ + fˆ dµ fˆ dη + fˆ dη fˆdη H n H H n H K n H K n H K H K H ≤ · (cid:48) · (cid:48) C b/Kδ + (cid:15)/3 + C b/Kδ < 3((cid:15)/3) = (cid:15), (cid:4) and we are done. Remark 3.1. Some similar ideas appear in the proof of Theorem 2.4 in [EMS03]. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: fˆdµ

text: fˆ

text: dµ

text: fˆdµ

text: C

text: dµ

text: H

text: H

text: K

text: H\

text: H\

text: CK

text: CK

text: C

text: dµ,

text: Kδ

text: H\

text: CK

text: where

text: we

text: choose

text: δ

text: such

text: that

text: δ

text: δ

text: We

text: introduce

text: this

text: extra

text: δ

text: to

text: get

text: the

text: term

text: Kδ

text: in

text: the

text: above,

text: which

text: will

text: allow

text: us

text: to

text: get

text: decay

text: as

text: K

text: By

text: Lemma

text: the

text: last

text: integral

text: above

text: is

text: ﬁnite;

text: we

text: need

text: to

text: show

text: the

text: somewhat

text: stronger

text: statement

text: that

text: it

text: is

text: bounded

text: from

text: above

text: independent

text: of

text: the

text: choice

text: of

text: µ.

text: The

text: idea

text: is

text: to

text: replace

text: the

text: integral

text: of

text: over

text: the

text: whole

text: stratum

text: by

text: the

text: integral

text: over

text: large

text: circles

text: centered

text: at

text: a

text: µ-generic

text: point,

text: using

text: Nevo’s

text: theorem,

text: and

text: then

text: use

text: Proposition

text: to

text: bound

text: the

text: integrals

text: over

text: circles.

text: To

text: apply

text: Nevo’s

text: theorem,

text: we

text: need

text: to

text: choose

text: a

text: smoothing

text: function

text: φ

text: R

text: R

text: that

text: is

text: non-negative,

text: smooth,

text: and

text: compactly

text: supported

text: (having

text: the

text: smoothing

text: is

text: ﬁne

text: for

text: our

text: purposes;

text: Nevo’s

text: result

text: should

text: also

text: be

text: true

text: without

text: having

text: to

text: smooth).

text: H

text: Now

text: by

text: Nevo’s

text: theorem

text: (see

text: Theorem

text: and

text: Proposition

text: for

text: µ

text: a.e.

text: X

text: 2π

text: dµ

text: φ(t)dt

text: l→im∞

text: φ(τ

text: t)

text: dθ

text: dt

text: H

text: τ

text: 2π

text: b

text: φ(t)dt.

text: Hence

text: H

text: dµ

text: b,

text: for

text: any

text: for

text: all

text: µ,

text: b

text: H

text: fˆdµ

text: H

text: fˆKdµ

text: C

text: Now

text: we

text: put

text: everything

text: together.

text: Fix

text: Choose

text: K

text: large

text: so

text: that

text: C

text: b/Kδ

text: Now

text: choose

text: N

text: such

text: that

text: fˆ

text: dµ

text: fˆ

text: dη

text: H

text: K

text: n

text: H

text: K

text: all

text: n

text: by

text: the

text: and

text: n

text: N,

text: fˆdµ

text: fˆdη

text: fˆdµ

text: fˆ

text: dµ

text: fˆ

text: dµ

text: fˆ

text: dη

text: fˆ

text: dη

text: fˆdη

text: H

text: n

text: H

text: H

text: n

text: H

text: K

text: n

text: H

text: K

text: n

text: H

text: K

text: H

text: K

text: H

text: C

text: b/Kδ

text: C

text: b/Kδ

text: and

text: we

text: are

text: done.

text: Remark

text: Some

text: similar

text: ideas

text: appear

text: in

text: the

text: proof

text: of

text: Theorem

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: BOUNDARY CONDITIONS AND MIXING AT DISCONTINUOUS PHASE TRANSITIONS 13 H ∩ + Λ • • b,φ n,n b b a S m h b,h,φ S b,m,φ 0 • φ • Figure 2. Left: the sets considered in Proposition 3.7, where we bound ± the probability of the interface exceeding height h. Right: the proof of Lemma 3.8 uses monotonicity in b.c., to go from bounding the prob- ability of the order-disorder interface exceeding height a in the gray set to the union of the gray and purple sets, to all shaded regions. The application of the FKG inequality here is valid as under (1, 0, b, φ) boundary I conditions, events of the sort “ exceeds some height a”, are decreasing events, as adding edges only pushes the order-disorder interface down. This property will be used throughout the paper. By monotonicity in boundary conditions again, I (cid:54)⊂ H− ∩ H ≤ I (cid:54)⊂ H− ∩ H π1,(cid:48)0,b,φ( + ) πH1,0,b,φ( + ) . S b+a,φ b,φ + b+a,φ b,φ b,φ By Proposition 3.6, there exists a c(δ, q) > 0 such that I (cid:54)⊂ H− ∩ H (cid:46) − (cid:4) πH1,0,b,φ( + ) n2e ca2/n . + b+a,φ b,φ b,φ ⊂ Z Proof of Proposition 3.7. As before, let (1, 0, b, φ) boundary conditions on D 2 ∩ H ∩ H− denote those that are wired on ∂D + and free on ∂D . By a union b+1,φ b+1,φ bound, write I (cid:54)⊂ S ≤ I (cid:54)⊂ H I (cid:54)⊂ H− πS1,0,b,φ∩ ( ) πS1,0,b,φ∩ ( +− ) + πS1,0,b,φ∩ ( ) (3.4) b,m,φ Λn,n b,h,φ b,m,φ Λn,n b h,φ b,m,φ Λn,n b+h,φ and consider the two quantities independently. Observe that the ﬁrst event on the right-hand side is an increasing event, while the second event is a decreasing event. By reﬂection symmetry and self-duality, if we prove the desired bound on the latter, for ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BOUNDARY

text: CONDITIONS

text: AND

text: MIXING

text: AT

text: DISCONTINUOUS

text: PHASE

text: TRANSITIONS

text: H

text: Λ

text: b,φ

text: n,n

text: b

text: b

text: a

text: S

text: m

text: h

text: b,h,φ

text: S

text: b,m,φ

text: φ

text: Figure

text: Left:

text: the

text: sets

text: considered

text: in

text: Proposition

text: where

text: we

text: bound

text: the

text: probability

text: of

text: the

text: interface

text: exceeding

text: height

text: h.

text: Right:

text: the

text: proof

text: of

text: Lemma

text: uses

text: monotonicity

text: in

text: b.c.,

text: to

text: go

text: from

text: bounding

text: the

text: prob-

text: ability

text: of

text: the

text: order-disorder

text: interface

text: exceeding

text: height

text: a

text: in

text: the

text: gray

text: set

text: to

text: the

text: union

text: of

text: the

text: gray

text: and

text: purple

text: sets,

text: to

text: all

text: shaded

text: regions.

text: The

text: application

text: of

text: the

text: FKG

text: inequality

text: here

text: is

text: valid

text: as

text: under

text: b,

text: φ)

text: boundary

text: I

text: conditions,

text: events

text: of

text: the

text: sort

text: exceeds

text: some

text: height

text: a”,

text: are

text: decreasing

text: events,

text: as

text: adding

text: edges

text: only

text: pushes

text: the

text: order-disorder

text: interface

text: down.

text: This

text: property

text: will

text: be

text: used

text: throughout

text: the

text: paper.

text: By

text: monotonicity

text: in

text: boundary

text: conditions

text: again,

text: I

text: H−

text: H

text: I

text: H−

text: H

text: S

text: b+a,φ

text: b,φ

text: b+a,φ

text: b,φ

text: b,φ

text: By

text: Proposition

text: there

text: exists

text: a

text: c(δ,

text: q)

text: such

text: that

text: I

text: H−

text: H

text: b+a,φ

text: b,φ

text: b,φ

text: Z

text: Proof

text: of

text: Proposition

text: As

text: before,

text: let

text: b,

text: φ)

text: boundary

text: conditions

text: on

text: D

text: H

text: H−

text: denote

text: those

text: that

text: are

text: wired

text: on

text: ∂D

text: and

text: free

text: on

text: ∂D

text: By

text: a

text: union

text: bound,

text: write

text: I

text: S

text: I

text: H

text: I

text: H−

text: b,m,φ

text: Λn,n

text: b,h,φ

text: b,m,φ

text: Λn,n

text: b

text: h,φ

text: b,m,φ

text: Λn,n

text: b+h,φ

text: and

text: consider

text: the

text: two

text: quantities

text: independently.

text: Observe

text: that

text: the

text: ﬁrst

text: event

text: on

text: the

text: right-hand

text: side

text: is

text: an

text: increasing

text: event,

text: while

text: the

text: second

text: event

text: is

text: a

text: decreasing

text: event.

text: By

text: reﬂection

text: symmetry

text: and

text: self-duality,

text: if

text: we

text: prove

text: the

text: desired

text: bound

text: on

text: the

text: latter,

text: for

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: BOUNDARY CONDITIONS AND MIXING AT DISCONTINUOUS PHASE TRANSITIONS 23 (cid:54) if X¯t1 = X¯t0, reset all edges in Λ to be wired/free. Let T = t2N+1 exp(n12+2ε); by monotonicity of FK Glauber dynamics, Theorem 2.7, and (2.6), it suﬃces to show P (cid:54) − (X¯1 = X¯0 ) = o(n 2) . (cid:110) T T (cid:111) (cid:91) In analogy to the proof of Theorem 2, let (cid:22) (cid:54) (cid:22) A = X0 − − = X1 − − where R = B . 2i t2i R2i B2i+1 B2i+2 t2i R2i B2i+1 B2i+2 i ≤ j j i Note that every time increment of t we make an independent attempt at coupling 2N+1 X¯1 to X¯0, and once they are coupled, in the corresponding chain that didn’t reset all t t edges to wired/free, they will be coupled for all subsequent times. Thus, it suﬃces to show that for large enough K, there exists c(δ, q) > 0, P(X¯t02N+1 = X¯t12N+1) ≥ P(Ac2N)P(Ac2N+1 | Ac2N) (cid:38) exp(−cn12+ε) . (4.4) as the probability of not coupling X¯T0 and X¯T1 by time T = t2N+1 exp(n12+2ε) would then be given by P(Bin(en12+2ε, e−cn12+ε) = 0) = o(n−2). In order to lower bound the probability in (4.4), we will prove the following inductively. Claim 4.6. There exists c(δ, q) > 0 such that, for large K and c , the following holds. 1 ≤ ≤ (1) For every 1 i N, there exists an event F measurable w.r.t. (X¯0, X¯1) ≤ such P (cid:22) ∈ · | i t t t t2i that (X¯0 ∪ F ) dominates the “wired” distribution on boundary t2i ∂w(B2i+1 B2i+∪2) i conditions o≤n ∂w≤(B2i+1 B2i+2). (2) For every 1 i N, the above deﬁned event F satisﬁes i P (cid:38) − (F , Ac ) exp( cin2ε) . i 2i Proof of Claim 4.6. For the base case of (1), observe that the boundary conditions ∪ on ∂w(B1 B2) dominate “wired” as they are all-wired; the base case of (2) follows as the proof of the inductive step does, so we do not repeat it here. Now assume that − items (1) and (2) hold for i 1 (for c(q) > 0 to be determined later) and show that they hold for i for the same choice of c. Consider the middle rectangle (cid:74) (cid:75) (cid:74) (cid:75) − × (cid:98) (cid:99) − (cid:98) (cid:99) (cid:110) D = (i 1)(cid:96), (i(cid:111)+ 1)(cid:96) n δn, n + δn , i 2 2 (cid:8) 2 (cid:9) and deﬁne the events (cid:22) I ∩ ∅ (cid:22) I ∩ − ∅ Γ − = X¯0 : D = , Γ = X¯0 : (B B ) = , 2i 1 t2i−1 B2i−1 i 2i t2i B2i 2i 2i+2 I (cid:22) where is the interface exposed by the boundary of the cluster(s) of X¯0 so that tj ∂wBj Γ − and Γ are increasing events. Then, we can lower bound 2i 1 2i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BOUNDARY

text: CONDITIONS

text: AND

text: MIXING

text: AT

text: DISCONTINUOUS

text: PHASE

text: TRANSITIONS

text: if

text: reset

text: all

text: edges

text: in

text: Λ

text: to

text: be

text: wired/free.

text: Let

text: T

text: by

text: monotonicity

text: of

text: FK

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text: SearchSnippets StackOverﬂow Biomedical Method NMI (%) NMI (%) NMI (%) ± ± ± K-means (TF) 09.03 2.30 07.81 2.56 09.36 2.04 ± ± ± K-means (TF-IDF) 21.40 4.35 15.64 4.68 25.43 3.23 ± ± ± SkipVec (Uni) 10.98 0.93 02.24 0.13 10.52 0.41 ± ± ± SkipVec (Bi) 09.27 0.29 02.89 0.20 10.15 0.59 ± ± ± SkipVec (Combine) 13.85 0.78 02.72 0.34 10.72 0.46 ± ± ± RecNN (Top) 04.04 0.74 09.90 0.96 08.87 0.53 ± ± ± RecNN (Ave.) 50.55 1.71 40.58 0.91 33.85 0.50 ± ± ± RecNN (Top+Ave.) 50.44 1.84 40.21 1.18 33.75 0.50 ± ± ± Para2vec 50.51 0.86 27.86 0.56 34.83 0.43 ± ± ± STC2-AE 54.01 1.55 38.22 1.31 33.58 0.48 ± ± ± STC2-LSA 54.53 1.47 34.38 1.12 33.90 0.67 ± ± ± STC2-LE 63.16 1.56 49.03 1.46 38.05 0.48 ± ± ± STC2-LPI 62.94 1.65 49.08 1.49 38.18 0.47 Table 5: Comparison of NMI of our proposed methods and three clustering methods on three datasets. For RecNN (Top), K-means is conducted on the learned vectors of the top tree node. For RecNN (Ave.), K-means is conducted on the average of all vectors in the tree. More details about the baseline setting are described in Section c that (1) BoW based approaches, including K-means (TF) and K-means (TF- IDF), and SkipVec based approaches perform not well; (2) RecNN based ap- proaches, both RecNN (Ave.) and RecNN (Top+Ave.), do better; (3) Para2vec makes a comparable performance with the most baselines; and (4) the evalua- tion clearly demonstrate the superiority of our proposed methods STC2. It is an expected results. For SkipVec based approaches, the oﬀ-the-shelf encoders are trained on the BookCorpus datasets [52], and then applied to our datasets to extract the sentence representations. The SkipVec encoders can produce generic sentence representations but may not perform well for speciﬁc datasets, in our experiments, StackOverﬂow and Biomedical datasets consist of many computer 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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label: table

text: StackOverﬂow

text: Biomedical

text: Method

text: NMI

text: NMI

text: NMI

text: K-means

text: (TF)

text: K-means

text: (TF-IDF)

text: SkipVec

text: (Uni)

text: SkipVec

text: (Bi)

text: SkipVec

text: (Combine)

text: RecNN

text: (Top)

text: RecNN

text: (Ave.)

text: RecNN

text: (Top+Ave.)

text: Table

text: Comparison

text: of

text: NMI

text: of

text: our

text: proposed

text: methods

text: and

text: three

text: clustering

text: methods

text: on

text: three

text: datasets.

text: For

text: RecNN

text: (Top),

text: K-means

text: is

text: conducted

text: on

text: the

text: learned

text: vectors

text: of

text: the

text: top

text: tree

text: node.

text: For

text: RecNN

text: (Ave.),

text: K-means

text: is

text: conducted

text: on

text: the

text: average

text: of

text: all

text: vectors

text: in

text: the

text: tree.

text: More

text: details

text: about

text: the

text: baseline

text: setting

text: are

text: described

text: in

text: Section

text: c

text: that

text: BoW

text: based

text: approaches,

text: including

text: K-means

text: (TF)

text: and

text: K-means

text: (TF-

text: IDF),

text: and

text: SkipVec

text: based

text: approaches

text: perform

text: not

text: well;

text: RecNN

text: based

text: ap-

text: proaches,

text: both

text: RecNN

text: (Ave.)

text: and

text: RecNN

text: (Top+Ave.),

text: do

text: better;

text: makes

text: a

text: comparable

text: performance

text: with

text: the

text: most

text: baselines;

text: and

text: the

text: evalua-

text: tion

text: clearly

text: demonstrate

text: the

text: superiority

text: of

text: our

text: proposed

text: methods

text: It

text: is

text: an

text: expected

text: results.

text: For

text: SkipVec

text: based

text: approaches,

text: the

text: oﬀ-the-shelf

text: encoders

text: are

text: trained

text: on

text: the

text: BookCorpus

text: datasets

text: and

text: then

text: applied

text: to

text: our

text: datasets

text: to

text: extract

text: the

text: sentence

text: representations.

text: The

text: SkipVec

text: encoders

text: can

text: produce

text: generic

text: sentence

text: representations

text: but

text: may

text: not

text: perform

text: well

text: for

text: speciﬁc

text: datasets,

text: in

text: our

text: experiments,

text: StackOverﬂow

text: and

text: Biomedical

text: datasets

text: consist

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text: SearchSnippets StackOverﬂow Biomedical Method NMI (%) NMI (%) NMI (%) ± ± ± bi-LSTM (last) 50.32 1.15 41.89 0.90 34.51 0.34 ± ± ± bi-LSTM (mean) 52.11 1.69 40.93 0.91 34.03 0.28 ± ± ± bi-LSTM (max) 46.81 2.38 36.73 0.56 31.90 0.23 ± ± ± bi-GRU (last) 56.00 0.75 38.73 0.78 32.91 0.40 ± ± ± bi-GRU (mean) 55.76 0.85 39.84 0.94 34.27 0.27 ± ± ± bi-GRU (max) 51.11 1.06 51.10 1.31 32.74 0.34 ± ± ± LPI (best) 38.48 2.39 27.21 0.88 29.73 0.30 ± ± ± STC2-LPI 62.94 1.65 49.08 1.49 38.18 0.47 Table 7: Comparison of NMI of our proposed methods and some other non-biased models on three datasets. For LPI, we project the text under the best dimension as described in Section c. For both bi-LSTM and bi-GRU based clustering methods, the binary codes generated from LPI are used to guide the learning of bi-LSTM/bi-GRU models. the results, we can see that bi-GRU and bi-LSTM based clustering methods do equally well, no clear winner, and both achieve great enhancements compared with LPI (best). Compared with these bi-LSTM/bi-GRU based models, the evaluation results still demonstrate the superiority of our approach methods, CNN based clustering model, in the most cases. As the results reported by Visin et al. [34], despite bi-directional or multi-directional RNN models perform a good non-biased feature extraction, they yet do not outperform state-of-the- art CNN on some tasks. In order to make clear what factors make our proposed method work, we report the bar chart results of ACC and MNI of our proposed methods and the corresponding baseline methods in Figure 3 and Figure 4. It is clear that, although AE and LSA does well or even better than LE and LPI, especially in dataset of both StackOverﬂow and Biomedical, STC2-LE and STC2-LPI achieve a much larger performance enhancements than STC2-AE and STC2- LSA do. The possible reason is that the information the pseudo supervision 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: SearchSnippets

text: StackOverﬂow

text: Biomedical

text: Method

text: NMI

text: NMI

text: NMI

text: bi-LSTM

text: (last)

text: bi-LSTM

text: (mean)

text: bi-LSTM

text: (max)

text: bi-GRU

text: (last)

text: bi-GRU

text: (mean)

text: bi-GRU

text: (max)

text: LPI

text: (best)

text: Table

text: Comparison

text: of

text: NMI

text: of

text: our

text: proposed

text: methods

text: and

text: some

text: other

text: non-biased

text: models

text: on

text: three

text: datasets.

text: For

text: LPI,

text: we

text: project

text: the

text: text

text: under

text: the

text: best

text: dimension

text: as

text: described

text: in

text: Section

text: c.

text: For

text: both

text: bi-LSTM

text: and

text: bi-GRU

text: based

text: clustering

text: methods,

text: the

text: binary

text: codes

text: generated

text: from

text: LPI

text: are

text: used

text: to

text: guide

text: the

text: learning

text: of

text: bi-LSTM/bi-GRU

text: models.

text: the

text: results,

text: we

text: can

text: see

text: that

text: bi-GRU

text: and

text: bi-LSTM

text: based

text: clustering

text: methods

text: do

text: equally

text: well,

text: no

text: clear

text: winner,

text: and

text: both

text: achieve

text: great

text: enhancements

text: compared

text: with

text: LPI

text: (best).

text: Compared

text: with

text: these

text: bi-LSTM/bi-GRU

text: based

text: models,

text: the

text: evaluation

text: results

text: still

text: demonstrate

text: the

text: superiority

text: of

text: our

text: approach

text: methods,

text: CNN

text: based

text: clustering

text: model,

text: in

text: the

text: most

text: cases.

text: As

text: the

text: results

text: reported

text: by

text: Visin

text: et

text: al.

text: despite

text: bi-directional

text: or

text: multi-directional

text: RNN

text: models

text: perform

text: a

text: good

text: non-biased

text: feature

text: extraction,

text: they

text: yet

text: do

text: not

text: outperform

text: state-of-the-

text: art

text: CNN

text: on

text: some

text: tasks.

text: In

text: order

text: to

text: make

text: clear

text: what

text: factors

text: make

text: our

text: proposed

text: method

text: work,

text: we

text: report

text: the

text: bar

text: chart

text: results

text: of

text: ACC

text: and

text: MNI

text: of

text: our

text: proposed

text: methods

text: and

text: the

text: corresponding

text: baseline

text: methods

text: in

text: Figure

text: and

text: Figure

text: It

text: is

text: clear

text: that,

text: although

text: AE

text: and

text: LSA

text: does

text: well

text: or

text: even

text: better

text: than

text: LE

text: and

text: LPI,

text: especially

text: in

text: dataset

text: of

text: both

text: StackOverﬂow

text: and

text: Biomedical,

text: and

text: achieve

text: a

text: much

text: larger

text: performance

text: enhancements

text: than

text: and

text: LSA

text: do.

text: The

text: possible

text: reason

text: is

text: that

text: the

text: information

text: the

text: pseudo

text: supervision

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text: Figure 4: NMI results on three short text datasets using our proposed STC2 based on AE, LSA, LE and LPI. in CNN is not departed very largely from each other, and the performance en- hancements of STC2-AE is also not quite satisfactory. For LE and LPI case, as we known that LE extracts the manifold structure of the original feature space, and LPI extracts both geometric and discriminating structure of the original fea- ture space [39]. We guess that our approach STC2-LE and STC2-LPI achieve enhancements compared with both LE and LPI by a large margin, because both of LE and LPI get useful semantic features, and these features are also diﬀerent from word embeddings used as input of CNN. From this view, we say that our proposed STC has potential to behave more eﬀective when the pseudo supervi- sion is able to get semantic meaningful features, which is diﬀerent enough from 24 ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: NMI

text: results

text: on

text: three

text: short

text: text

text: datasets

text: using

text: our

text: proposed

text: based

text: on

text: AE,

text: LSA,

text: LE

text: and

text: LPI.

text: in

text: CNN

text: is

text: not

text: departed

text: very

text: largely

text: from

text: each

text: other,

text: and

text: the

text: performance

text: en-

text: hancements

text: of

text: is

text: also

text: not

text: quite

text: satisfactory.

text: For

text: LE

text: and

text: LPI

text: case,

text: as

text: we

text: known

text: that

text: LE

text: extracts

text: the

text: manifold

text: structure

text: of

text: the

text: original

text: feature

text: space,

text: and

text: LPI

text: extracts

text: both

text: geometric

text: and

text: discriminating

text: structure

text: of

text: the

text: original

text: fea-

text: ture

text: space

text: We

text: guess

text: that

text: our

text: approach

text: and

text: achieve

text: enhancements

text: compared

text: with

text: both

text: LE

text: and

text: LPI

text: by

text: a

text: large

text: margin,

text: because

text: both

text: of

text: LE

text: and

text: LPI

text: get

text: useful

text: semantic

text: features,

text: and

text: these

text: features

text: are

text: also

text: diﬀerent

text: from

text: word

text: embeddings

text: used

text: as

text: input

text: of

text: CNN.

text: From

text: this

text: view,

text: we

text: say

text: that

text: our

text: proposed

text: STC

text: has

text: potential

text: to

text: behave

text: more

text: eﬀective

text: when

text: the

text: pseudo

text: supervi-

text: sion

text: is

text: able

text: to

text: get

text: semantic

text: meaningful

text: features,

text: which

text: is

text: diﬀerent

text: enough

text: from

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Figure 2: The architecture of dynamic convolutional neural network [11]. An input text is ﬁrst projected to a matrix feature by looking up word embeddings, and then goes through wide convolutional layers, folding layers and k-max pooling layers, which provides a deep feature representation before the output layer. last section describe how to perform clustering on the learned features. 3.1. Deep Convolutional Neural Networks In this section, we brieﬂy review one popular deep convolutional neural net- work, Dynamic Convolutional Neural Network (DCNN) [11] as an instance of CNN in the following sections, which as the foundation of our proposed method has been successfully proposed for the completely supervised learning task, text classiﬁcation. Taking a neural network with two convolutional layers in Figure 2 as an example, the network transforms raw input text to a powerful representation. Particularly, each raw text vector x is projected into a matrix representation i ∈ R × S dw s by looking up a word embedding E, where s is the length of one { } text. We also let W˜ = W and W denote the weights of the neural i i=1,2 O · R × → R × (cid:29) networks. The network deﬁnes a transformation f( ) : d 1 r 1 (d r) which transforms an input raw text x to a r-dimensional deep representation h. There are three basic operations described as follows: 8 ##LTFigure##

text: Figure

text: The

text: architecture

text: of

text: dynamic

text: convolutional

text: neural

text: network

text: An

text: input

text: text

text: is

text: ﬁrst

text: projected

text: to

text: a

text: matrix

text: feature

text: by

text: looking

text: up

text: word

text: embeddings,

text: and

text: then

text: goes

text: through

text: wide

text: convolutional

text: layers,

text: folding

text: layers

text: and

text: k-max

text: pooling

text: layers,

text: which

text: provides

text: a

text: deep

text: feature

text: representation

text: before

text: the

text: output

text: layer.

text: last

text: section

text: describe

text: how

text: to

text: perform

text: clustering

text: on

text: the

text: learned

text: features.

label: section

text: Deep

text: Convolutional

text: Neural

text: Networks

text: In

text: this

text: section,

text: we

text: brieﬂy

text: review

text: one

text: popular

text: deep

text: convolutional

text: neural

text: net-

text: work,

text: Dynamic

text: Convolutional

text: Neural

text: Network

text: (DCNN)

text: as

text: an

text: instance

text: of

text: CNN

text: in

text: the

text: following

text: sections,

text: which

text: as

text: the

text: foundation

text: of

text: our

text: proposed

text: method

text: has

text: been

text: successfully

text: proposed

text: for

text: the

text: completely

text: supervised

text: learning

text: task,

text: text

text: classiﬁcation.

text: Taking

text: a

text: neural

text: network

text: with

text: two

text: convolutional

text: layers

text: in

text: Figure

text: as

text: an

text: example,

text: the

text: network

text: transforms

text: raw

text: input

text: text

text: to

text: a

text: powerful

text: representation.

text: Particularly,

text: each

text: raw

text: text

text: vector

text: x

text: is

text: projected

text: into

text: a

text: matrix

text: representation

text: i

text: R

text: S

text: dw

text: s

text: by

text: looking

text: up

text: a

text: word

text: embedding

text: E,

text: where

text: s

text: is

text: the

text: length

text: of

text: one

text: text.

text: We

text: also

text: let

text: W˜

text: W

text: and

text: W

text: denote

text: the

text: weights

text: of

text: the

text: neural

text: i

text: O

text: R

text: R

text: networks.

text: The

text: network

text: deﬁnes

text: a

text: transformation

text: f(

text: d

text: r

text: (d

text: r)

text: which

text: transforms

text: an

text: input

text: raw

text: text

text: x

text: to

text: a

text: r-dimensional

text: deep

text: representation

text: h.

text: There

text: are

text: three

text: basic

text: operations

text: described

text: as

text: follows:

text: ##LTFigure##


text: stable for any jamming burstiness, unlike the full sensing algorithms we considered, which have in their codes a bound on jamming burstiness they can handle in a stable manner. Lemma 2 Consider an execution of algorithm OFC-RRW against a leaky-bucket adversary of some type (ρ, λ, b). If there are x old packets in the system in some round, then at least x packets are transmitted within the next x+−n+b rounds. 1 λ Proof: It takes n control rounds for the token to pass through all n stations. Consider a contiguous time segment of y rounds in which some x packets are heard. At most yλ + b of these y rounds can be jammed. It follows that the following inequality holds: ≤ y n + x + yλ + b . Solving for y yields the upper bound ≤ x + n + b y − , 1 λ (cid:3) as the bound on the length of a contiguous time interval in whi(cid:0)ch at le(cid:1)ast x packets are heard. O Theorem 3 The packet latency of algorithm OFC-RRW is −n+−b when executed against the 1 ρ λ jamming adversary of type (ρ, λ, b). Proof: Let t denote the duration of phase i and q be the number of old packets in the beginning i ≥ i of phase i, for i 1. We use the following two estimates to derive a recurrence for the numbers t . i One is ≤ q ρt + b , (8) i+1 i which follows from the deﬁnition of old packets and the adversary of type (ρ, λ, b). The other is ≤ q + n + b t i+1 − , (9) i+1 1 λ which follows from Lemma 2. Using the abbreviations c = n+−2b and d = −ρ , we substitute (8) 1 λ 1 λ into (9) to obtain ≤ n + b ρt + b t − + i− i+1 1 λ 1 λ n + 2b ρ = − + − t i 1 λ 1 λ ≤ c + dt . i ≤ To ﬁnd an upper bound on the duration of a phase, we iterate the recurrence t c + dt , i+1 i which produces 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: stable

text: for

text: any

text: jamming

text: burstiness,

text: unlike

text: the

text: full

text: sensing

text: algorithms

text: we

text: considered,

text: which

text: have

text: in

text: their

text: codes

text: a

text: bound

text: on

text: jamming

text: burstiness

text: they

text: can

text: handle

text: in

text: a

text: stable

text: manner.

text: Lemma

text: Consider

text: an

text: execution

text: of

text: algorithm

text: OFC-RRW

text: against

text: a

text: leaky-bucket

text: adversary

text: of

text: some

text: type

text: (ρ,

text: λ,

text: b).

text: If

text: there

text: are

text: x

text: old

text: packets

text: in

text: the

text: system

text: in

text: some

text: round,

text: then

text: at

text: least

text: x

text: packets

text: are

text: transmitted

text: within

text: the

text: next

text: x+−n+b

text: rounds.

text: λ

text: Proof:

text: It

text: takes

text: n

text: control

text: rounds

text: for

text: the

text: token

text: to

text: pass

text: through

text: all

text: n

text: stations.

text: Consider

text: a

text: contiguous

text: time

text: segment

text: of

text: y

text: rounds

text: in

text: which

text: some

text: x

text: packets

text: are

text: heard.

text: At

text: most

text: yλ

text: b

text: of

text: these

text: y

text: rounds

text: can

text: be

text: jammed.

text: It

text: follows

text: that

text: the

text: following

text: inequality

text: holds:

text: y

text: n

text: x

text: yλ

text: b

text: Solving

text: for

text: y

text: yields

text: the

text: upper

text: bound

text: x

text: n

text: b

text: y

text: λ

text: as

text: the

text: bound

text: on

text: the

text: length

text: of

text: a

text: contiguous

text: time

text: interval

text: in

text: at

text: x

text: packets

text: are

text: heard.

text: O

text: Theorem

text: The

text: packet

text: latency

text: of

text: algorithm

text: OFC-RRW

text: is

text: −n+−b

text: when

text: executed

text: against

text: the

text: ρ

text: λ

text: jamming

text: adversary

text: of

text: type

text: (ρ,

text: λ,

text: b).

text: Proof:

text: Let

text: t

text: denote

text: the

text: duration

text: of

text: phase

text: i

text: and

text: q

text: be

text: the

text: number

text: of

text: old

text: packets

text: in

text: the

text: beginning

text: i

text: i

text: of

text: phase

text: i,

text: for

text: i

text: We

text: use

text: the

text: following

text: two

text: estimates

text: to

text: derive

text: a

text: recurrence

text: for

text: the

text: numbers

text: t

text: i

text: One

text: is

text: q

text: ρt

text: b

text: i

text: which

text: follows

text: from

text: the

text: deﬁnition

text: of

text: old

text: packets

text: and

text: the

text: adversary

text: of

text: type

text: (ρ,

text: λ,

text: b).

text: The

text: other

text: is

text: q

text: n

text: b

text: t

text: λ

text: which

text: follows

text: from

text: Lemma

text: Using

text: the

text: abbreviations

text: c

text: and

text: d

text: −ρ

text: we

text: substitute

text: λ

text: λ

text: into

text: to

text: obtain

text: n

text: b

text: ρt

text: b

text: t

text: i−

text: λ

text: λ

text: n

text: ρ

text: t

text: i

text: λ

text: λ

text: c

text: dt

text: i

text: To

text: ﬁnd

text: an

text: upper

text: bound

text: on

text: the

text: duration

text: of

text: a

text: phase,

text: we

text: iterate

text: the

text: recurrence

text: t

text: c

text: dt

text: i

text: which

text: produces

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≤ ≤ c t c + dc + d2c + . . . dic − . (10) i+1 1 d After substituting c = n+−2b and d = −ρ into (3), we obtain, by algebraic manipulations, the 1 λ 1 λ following estimate: ≤ n + 2b · 1 t − − i 1 λ 1 −ρ −1 λ ≤ n + 2b · 1 λ − − − 1 λ 1 ρ λ ≤ 2(n + b) − − . (11) 1 ρ λ The bound on packet latency we seek is twice that in (11), because a packet is queued for at (cid:3) (cid:0) (cid:1) most two consecutive phases. O Theorem 4 The packet latency of algorithm C-RRW is −n+−b , when executed against the (1 ρ λ)2 jamming adversary of type (ρ, λ, b). Proof: We compare packet latency of algorithm C-RRW to that of algorithm OFC-RRW. Consider executions of algorithms C-RRW and OFC-RRW for the same injection and jamming pattern of the adversary. Let s and t be the bounds on the length of phase i of algorithms C-RRW i i and OFC-RRW, respectively. Phase i of C-RRW takes s rounds. When algorithm OFC-RRW is executed, then its phase i i may take up to the following number of rounds: s s + s (ρ + λ) + s (ρ + λ)2 + . . . = − i . i i i 1 (ρ + λ) So the phase’s length of algorithm C-RRW diﬀers from that of algorithm OFC-RRW by a factor of at most −1− . An injected packet is transmitted in at most two phases of an execution, for 1 ρ λ each of the two considered algorithms. The bound of Theorem 3 is twice the bound on the duration of a phase. It follows that a bound on packet latency of algorithm C-RRW can be obtained by (cid:3) multiplying the bound given in Theorem 3 by −1− . 1 ρ λ (cid:0) (cid:1) The tightness of the bound on packet latency given in Theorem 4 can be established similarly as that for Theorem 2. To this end, replace n(J + 1) by n in the tightness argument for the bound of Theorem 2 given in Section to obtain Ω −n+−b as a bound. The tightness of the upper (1 ρ λ)2 bound of Theorem 3 follows from the property that the derivation of the upper bound can be closely mimicked by the adversary. Alternatively, we observe that an improvement of the bound in Theorem 3 would lead to an improvement to the bound in Theorem 4, which has already been shown to be tight. Algorithm Move-Big-To-Front. Next we estimate the packet latency of algorithm MBTF against jamming. The analysis we give resorts to estimates of the number of packets stored in the queues at all times. 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: c

text: t

text: c

text: dc

text: dic

text: d

text: After

text: substituting

text: c

text: and

text: d

text: −ρ

text: into

text: we

text: obtain,

text: by

text: algebraic

text: manipulations,

text: the

text: λ

text: λ

text: following

text: estimate:

text: n

text: t

text: i

text: λ

text: −ρ

text: −1

text: λ

text: n

text: λ

text: λ

text: ρ

text: λ

text: b)

text: ρ

text: λ

text: The

text: bound

text: on

text: packet

text: latency

text: we

text: seek

text: is

text: twice

text: that

text: in

text: because

text: a

text: packet

text: is

text: queued

text: for

text: at

text: most

text: two

text: consecutive

text: phases.

text: O

text: Theorem

text: The

text: packet

text: latency

text: of

text: algorithm

text: C-RRW

text: is

text: −n+−b

text: when

text: executed

text: against

text: the

text: ρ

text: jamming

text: adversary

text: of

text: type

text: (ρ,

text: λ,

text: b).

text: Proof:

text: We

text: compare

text: packet

text: latency

text: of

text: algorithm

text: C-RRW

text: to

text: that

text: of

text: algorithm

text: OFC-RRW.

text: Consider

text: executions

text: of

text: algorithms

text: C-RRW

text: and

text: OFC-RRW

text: for

text: the

text: same

text: injection

text: and

text: jamming

text: pattern

text: of

text: the

text: adversary.

text: Let

text: s

text: and

text: t

text: be

text: the

text: bounds

text: on

text: the

text: length

text: of

text: phase

text: i

text: of

text: algorithms

text: C-RRW

text: i

text: i

text: and

text: OFC-RRW,

text: respectively.

text: Phase

text: i

text: of

text: C-RRW

text: takes

text: s

text: rounds.

text: When

text: algorithm

text: OFC-RRW

text: is

text: executed,

text: then

text: its

text: phase

text: i

text: i

text: may

text: take

text: up

text: to

text: the

text: following

text: number

text: of

text: rounds:

text: s

text: s

text: s

text: (ρ

text: λ)

text: s

text: (ρ

text: i

text: i

text: i

text: i

text: (ρ

text: λ)

text: So

text: the

text: phase’s

text: length

text: of

text: algorithm

text: C-RRW

text: diﬀers

text: from

text: that

text: of

text: algorithm

text: OFC-RRW

text: by

text: a

text: factor

text: of

text: at

text: most

text: −1−

text: An

text: injected

text: packet

text: is

text: transmitted

text: in

text: at

text: most

text: two

text: phases

text: of

text: an

text: execution,

text: for

text: ρ

text: λ

text: each

text: of

text: the

text: two

text: considered

text: algorithms.

text: The

text: bound

text: of

text: Theorem

text: is

text: twice

text: the

text: bound

text: on

text: the

text: duration

text: of

text: a

text: phase.

text: It

text: follows

text: that

text: a

text: bound

text: on

text: packet

text: latency

text: of

text: algorithm

text: C-RRW

text: can

text: be

text: obtained

text: by

text: multiplying

text: the

text: bound

text: given

text: in

text: Theorem

text: by

text: −1−

text: ρ

text: λ

text: The

text: tightness

text: of

text: the

text: bound

text: on

text: packet

text: latency

text: given

text: in

text: Theorem

text: can

text: be

text: established

text: similarly

text: as

text: that

text: for

text: Theorem

text: To

text: this

text: end,

text: replace

text: n(J

text: by

text: n

text: in

text: the

text: tightness

text: argument

text: for

text: the

text: bound

text: of

text: Theorem

text: given

text: in

text: Section

text: to

text: obtain

text: Ω

text: −n+−b

text: as

text: a

text: bound.

text: The

text: tightness

text: of

text: the

text: upper

text: ρ

text: bound

text: of

text: Theorem

text: follows

text: from

text: the

text: property

text: that

text: the

text: derivation

text: of

text: the

text: upper

text: bound

text: can

text: be

text: closely

text: mimicked

text: by

text: the

text: adversary.

text: Alternatively,

text: we

text: observe

text: that

text: an

text: improvement

text: of

text: the

text: bound

text: in

text: Theorem

text: would

text: lead

text: to

text: an

text: improvement

text: to

text: the

text: bound

text: in

text: Theorem

text: which

text: has

text: already

text: been

text: shown

text: to

text: be

text: tight.

text: Algorithm

text: Move-Big-To-Front.

text: Next

text: we

text: estimate

text: the

text: packet

text: latency

text: of

text: algorithm

text: MBTF

text: against

text: jamming.

text: The

text: analysis

text: we

text: give

text: resorts

text: to

text: estimates

text: of

text: the

text: number

text: of

text: packets

text: stored

text: in

text: the

text: queues

text: at

text: all

text: times.

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##