Solving interacting particle systems by Fourier-like transforms - - PowerPoint PPT Presentation

Solving interacting particle systems by Fourier-like transforms

Leonid Petrov

University of Virginia

April 13, 2015

Stochastic higher spin vertex model

Stochastic higher spin vertex model (J = 1) particle interpretation: spin interpretation:

Stochastic higher spin vertex model (J = 1) particle interpretation: spin interpretation: Vertex weights (q, ν = q−I, α, J = 1)

g g 1 + αqg 1 + α g g − 1 1 α(1 − qg) 1 + α g 1 g + 1 1 − νqg 1 + α g 1 g 1 α + νqg 1 + α

Vertex weights (q, ν = q−I, α, J = 1)

g g 1 + αqg 1 + α g g − 1 1 α(1 − qg) 1 + α g 1 g + 1 1 − νqg 1 + α g 1 g 1 α + νqg 1 + α

Stochastic when q, ν ∈ [0, 1), and α ≥ 0, q ∈ (−1, 0], α ∈ (0, 1/|q|), and ν ∈

• − 1/|q|, min(1, α/|q|)
• ,

q ∈ [0, 1), ν = q−I for I ∈ Z≥1, and α < −q−I, q ∈ (1, +∞), ν = q−I for I ∈ Z≥1, and −q−I < α < 0. The two latter cases bound the number of allowed vertical spins by I ∈ Z≥1.

General higher spin vertex model (q, ν, α; J ∈ Z≥1) — obtained by fusion from the J = 1 case. Vertex weights are expressed through q-Racah univariate

• rthogonal polynomials, or basic hypergeometric functions 4φ3.

General higher spin vertex model (q, ν, α; J ∈ Z≥1) — obtained by fusion from the J = 1 case. One can have qJ ∈ C (because general J vertex weights are polynomial in qJ). The model is stochastic if, for example, J ∈ Z≥1;

• r α = −ν, qJα = −µ with 0 ≤ ν ≤ µ < 1:

q-Hahn process [Povolotsky ’13], [Corwin ’14] General J system degenerates to most of the known Bethe ansatz integrable (1+1)d models from the Kardar-Parisi-Zhang universality class. (there are other processes with duality not fitting into this scheme, cf. [Carinci, Giardina, Redig, Sasamoto ’14])

General higher spin vertex model (q, ν, α; J ∈ Z≥1) Degenerates to most of the known Bethe ansatz integrable (1+1)d models from the Kardar-Parisi-Zhang universality class.

Stochastic higher spin q-Hahn TASEP exclusion process / zero range process Discrete time q-TASEP q-TASEP Semi-discrete Brownian polymer Strict/weak Stochastic six-vertex model ASEP Brownian motions KPZ equation / SHE / continuum polymer KPZ fixed point (e.g. Tracy-Widom distributions) with skew reflection polymer

[Korepin, Bogoli- ubov, Izergin, ’97] (ABA) [Kirillov–Reshetikhin ’87] (fusion) [Mangazeev ’14] (R matrices) [Borodin ’14] [Corwin–P. ’15]

Examples

Stochastic six vertex model (q > 1; ν = 1/q; −1/q < α < 0; J = 1)

[Gwa-Spohn ’92], [Borodin–Corwin–Gorin ’14]

Stochastic six vertex model (q > 1; ν = 1/q; −1/q < α < 0; J = 1)

[Gwa-Spohn ’92], [Borodin–Corwin–Gorin ’14]

Infinite-spin model (q; 1 > ν > 0; α > 0; J = 1)

4 vertical, 3 horizontal (q; ν = q−4; α < −q−4; J = 3)

Eigenfunctions

Space reversed particle system Restrict the (q, ν, α, J) system to k-particle configurations (n1 ≥ n2 ≥ . . . ≥ nk). The transition operator (= transfer matrix) ˜ Bα,qJα. Eigenfunctions depend on z = (z1, . . . , zk) ∈ Ck: ˜ Bα,qJαΨℓ

• z
• (

n) =

k

• j=1

1 + qJαzj 1 + αzj Ψℓ

• z(

n)

Left eigenfunctions ˜ Bα,qJαΨℓ

• z
• (

n) =

k

• j=1

1 + qJαzj 1 + αzj Ψℓ

• z(

n), (under condition

• 1−zi

1−νzi qjα+ν 1+qjα

• < 1 for all i and 1 ≤ j ≤ J − 1),

where Ψℓ

• z(

n) =

• σ∈S(k)
• 1≤B<A≤k

zσ(A) − qzσ(B) zσ(A) − zσ(B)

k

• j=1

1 − zσ(j) 1 − νzσ(j) −nj . Remark The operators ˜ Bα,qJα depend on (q, ν, α, J), and eigenfunctions — only on (q, ν): commuting system of transfer matrices. Remark The eigenfunctions are “algebraic”: they are not compactly sup- ported in n (which is a natural domain for ˜ Bα,qJα).

• Cf. exponents as eigenfunctions of d/dx.

Left eigenfunctions ˜ Bα,qJαΨℓ

• z
• (

n) =

k

• j=1

1 + qJαzj 1 + αzj Ψℓ

• z(

n), where Ψℓ

• z(

n) =

• σ∈S(k)
• 1≤B<A≤k

zσ(A) − qzσ(B) zσ(A) − zσ(B)

k

• j=1

1 − zσ(j) 1 − νzσ(j) −nj . Right eigenfunctions Ψr

• z(

n) = Cq,ν( n)

• σ∈S(k)
• B<A

qzσ(A) − zσ(B) zσ(A) − zσ(B)

k

• j=1

1 − zσ(j) 1 − νzσ(j) nj , with

• Ψr
• z ˜

Bα,qJα ( n) =

k

• j=1

1 + qJαzj 1 + αzj Ψr

• z.

The operator ˜ Bα,qJα is not (Hermitian) symmetric, but is con- jugate to its transpose (PT-symmetry).

Solving particle systems Goal Understand large-time behavior of the system, i.e., raise the matrix (operator) ˜ Bα,qJα to a large power, and preferably be able to apply it to an arbitrary vector (i.e., arbitrary initial data). Via self-duality, this also gives moment information.

1 Diagonalize the operator (below: ASEP example) 2 Plancherel theory associated to eigenfunctions provides

mutually inverse Fourier-like transforms

3 Project the initial data to eigenfunctions, evolve in the

spectral space, then go back to the coordinate space using the inverse Fourier-like transform.

Bonuses: • Fredholm determinants for special initial data;

• explanation of Tracy-Widom’s symmetrization formulas
• theory of symmetric functions associated with eigenfunctions

(Cauchy and Pieri identities, etc.) [Borodin ’14]

Coordinate Bethe ansatz for ASEP

Coordinate Bethe ansatz for ASEP Let me explain the origin of the eigenfunctions in a simpler setup of the ASEP (first non-determinantal model shown to belong to the KPZ universality class [Tracy-Widom ’07+]).

x1 x2 x3 xk R L R

Let R + L = 1, q = R/L < 1, and H(k) be the Markov generator of the k-particle ASEP (in fact, it is conjugate to the XXZ Hamiltonian; the latter is not stochastic). Let the ASEP coordinates be x1 < x2 < . . . < xk.

Coordinate Bethe ansatz for ASEP k = 1 : H(1)f (x1) = R(f (x1 + 1) − f (x1)) + L(f (x1 − 1) − f (x1)).

Coordinate Bethe ansatz for ASEP k = 1 : H(1)f (x1) = R(f (x1 + 1) − f (x1)) + L(f (x1 − 1) − f (x1)). k = 2, x1 + 1 < x2: H(2)f (x1, x2) = R(f (x1 + 1, x2) − f (x1, x2)) + L(f (x1 − 1, x2) − f (x1, x2)) + R(f (x1, x2 + 1) − f (x1, x2)) + L(f (x1, x2 − 1) − f (x1, x2)) =

• H(1)

1

+ H(1)

2

• f (x1, x2).

Coordinate Bethe ansatz for ASEP k = 1 : H(1)f (x1) = R(f (x1 + 1) − f (x1)) + L(f (x1 − 1) − f (x1)). k = 2, x1 + 1 < x2: H(2)f (x1, x2) = R(f (x1 + 1, x2) − f (x1, x2)) + L(f (x1 − 1, x2) − f (x1, x2)) + R(f (x1, x2 + 1) − f (x1, x2)) + L(f (x1, x2 − 1) − f (x1, x2)) =

• H(1)

1

+ H(1)

2

• f (x1, x2).

k = 2, x1 + 1 = x2: x1 cannot jump right, x2 cannot jump left H(2)f (x1, x2) = R(f (x1, x2 + 1) − f (x1, x2)) + L(f (x1 − 1, x2) − f (x1, x2)) =

• H(1)

1

+ H(1)

2

• f (x1, x2) + discrepancy,

discrepancy = Rf (x1 + 1, x2) + Lf (x1, x2 − 1) − f (x1, x2)

Coordinate Bethe ansatz for ASEP When x1 + 1 = x2, discrepancy = Rf (x1 + 1, x2) + Lf (x1, x2 − 1) − f (x1, x2) involves values of f outside the “physical region” x1 < x2. Therefore, we can assign arbitrary values to f outside this region so that discrepancy = 0. Can do the same for k particles, and the boundary conditions will involve only pairs of neighboring particles (two-body boundary conditions). ASEP is integrable in the sense of [Bethe ’31] H(k)f =

• H(1)

1

+ . . . + H(1)

k

• f if f such that for any i,

Rf (. . . , xi +1, xi+1, . . .)+Lf (. . . , xi, xi+1 −1, . . .)−f (. . .) = 0 whenever xi + 1 = xi+1.

Coordinate Bethe ansatz for ASEP Therefore, one can diagonalize each H(1)

i

separately, and combine the eigenfunctions so that to satisfy the boundary conditions. The sum of one-particle operators has eigenfunctions

• σ∈S(k)

Aσ( z)

k

• i=1

1 + zσ(i) 1 + zσ(i)/q −xi ,

• z = (z1, . . . , zk) ∈ Ck.

These will be eigenfunctions for any choice of Aσ( z). Then it is possible to choose Aσ( z) to satisfy the boundary conditions, and thus one has ASEP eigenfunctions

• σ∈S(k)
• B<A

zσ(B) − qzσ(A) zσ(B) − zσ(A)

k

• i=1

1 + zσ(i) 1 + zσ(i)/q −xi

Coordinate Bethe ansatz for the higher spin vertex model

How to obtain the eigenfunctions of ˜ Bα,qJα? The operator ˜ Bα,qJα cannot be written as “free operator + boundary conditions” (like it was in the ASEP case), but instead ˜ Bα,qJα = ( ˜ Hα)−1( ˜ HqJα), where operators ˜ Hα commute for different α, and they can be written as “free operator + boundary conditions”. The operators ˜ Hα come from the q-Hahn particle system

[Povolotsky ’13], and

˜ HαΨℓ

• z
• (

n) =

k

• j=1

1 + αzj 1 − νzj Ψℓ

• z(

n).

How to obtain the eigenfunctions of ˜ Bα,qJα? ˜ HαΨℓ

• z
• (

n) =

k

• j=1

1 + αzj 1 − νzj Ψℓ

• z(

n). One-particle operator 1 + α 1 − ν f (n) − α + ν 1 − ν f (n − 1), and ˜ Hαf is equal to the product of one-particle operators (discrete time) if f satisfies certain two-body boundary conditions:

• ν(1 − q)f (

n−

i,i+1) + (q − ν)f (

n−

i+1)

+(1 − q)f ( n) − (1 − qν)f ( n−

i )

• n∈Zk : ni=ni+1

= 0 (coordinates are n1 ≥ n2 ≥ . . . ≥ nk now). Eigenfunctions of ˜ Hα were obtained in this way in [Povolotsky ’13]. Also related to affine Hecke algebras [Takeyama ’14]

Plancherel theory

Direct transform

Two spaces of functions

1 Space of compactly supported functions of the spatial

variables n

2 Space of symmetric Laurent polynomials in 1−zj 1−νzj

Direct transform of f ( n) (Fq,νf )( z) :=

• n

f ( n)Ψr

• z(

n) =:

• f , Ψr
• z
• n.

Inverse transform

Inverse transform of G( z) (J q,νG)( n) :=

• γ1

dz1 2πi . . .

• γk

dzk 2πi

• A<B

zA − zB zA − qzB ×

k

• j=1

1 (1 − zj)(1 − νzj) 1 − zj 1 − νzj −nj G( z).

1 q q2 ν−1 γ3 γ2 γ1

Can write the inverse transform as (J q,νG)( n) =

• G, Ψℓ(

n)

• z

Plancherel isomorphism theorems

Theorem [Borodin–Corwin–P.–Sasamoto ’14] (1) The composition J q,νFq,ν is an identity operator on the space of compactly supported functions in n. (2) The composition Fq,νJ q,ν is an identity operator on the space of symmetric Laurent polynomials in

1−zj 1−νzj .

Proof of (1) is a direct combinatorial argument with shrinking/expanding integration contours: show that

• J q,νΨr
• (

x)

• (

y) = 1

x=

• y. In fact, contribution from each

permutation σ in Ψr

• z vanishes individually if

x = y. Proof of (2) relies on the presence of a commuting family of

• perators ˜

Hα diagonalized in the same eigenfunctions. This gives many relations that we use.

Biorthogonality

Spatial biorthogonality

• Ψℓ
• (

x), Ψr

• (

y)

• z = 1

x= y.

Follows immediately from the first Plancherel theorem. Also this implies completeness of the Bethe ansatz.

Biorthogonality

Spatial biorthogonality

• Ψℓ
• (

x), Ψr

• (

y)

• z = 1

x= y.

Follows immediately from the first Plancherel theorem. Also this implies completeness of the Bethe ansatz. Spectral biorthogonality In a certain weak sense (V( z) is the Vandermonde),

• n

Ψr

• z(

n)Ψℓ

• w(

n)V( z)V( w) = (−1)

k(k−1) 2

k

• j=1

(1 − zj)(1 − νzj)

• A=B

(zA − qzB) det[δzi,wj]k

i,j=1.

This statement in fact implies the second Plancherel theorem.

Theory of symmetric functions [Borodin ’14] The functions Ψ•

• z(

n) have a rich algebraic structure (as symmetric rational functions). In fact, they degenerate to the Hall-Littlewood symmetric polynomials, and inherit many of their properties, such as the Cauchy identity:

• n: nk≥0

Ψr

• z(

n)Φ

w(

n) = Const ·

k

• i=1

k

• j=1

1 − qziwj 1 − ziwj , where Φ

w(

n) are certain “dual” functions (like the P and Q Hall-Littlewood polynomials, but now the Φ’s are not constant multiples of the Ψ’s). This Cauchy identity follows from the Yang-Baxter relation.

Theory of symmetric functions [Borodin ’14]

• n: nk≥0

Ψr

• z(

n)Φ

w(

n) = Const ·

k

• i=1

k

• j=1

1 − qziwj 1 − ziwj , where Φ

w(

n) are certain “dual” functions (like the P and Q Hall-Littlewood polynomials, but now the Φ’s are not constant multiples of the Ψ’s). This Cauchy identity follows from the Yang-Baxter relation. Moreover, Ψ and Φ are essentially partition functions of certain configurations of the J = 1 vertex model:

Moment formulas

Exclusion process associated with the vertex model Let I / ∈ Z≥1, and consider the process

• x = (x1 > x2 > . . . > xN) with gaps gi = xi − xi+1 − 1

evolving according to Bα,qJα (the vertex model with spins going to the right). In discrete time, each particle xi jumps by at most J, and update propagates from particle x1 leftwards.

Moment formulas for the exclusion process Step initial data xi(0) = −i, i ∈ Z≥1. Theorem [Corwin-P. ’15]. For all n1 ≥ · · · ≥ nk ≥ 1 and t ∈ Z≥0, E

• k
• i=1

qxni (t)+ni

• = (−1)kq

k(k−1) 2

(2πi)k

• γ1

· · ·

• γk
• A<B

zA − zB zA − qzB ×

k

• j=1

1 − zj 1 − νzj −nj1 + qJαzj 1 + αzj t dzj zj(1 − νzj) k

j=1

• 1+qJαzj

1+αzj

t is the eigenvalue, and factors

dzj zj(1−νzj) come

from pairing of the step initial data with the

• eigenfunctions. One can write similar formulas for an

arbitrary initial data, but the pairing is in general not explicit. The contour integration is a result of the application of the inverse Fourier transform.

Moment formulas for the exclusion process Step initial data xi(0) = −i, i ∈ Z≥1. Theorem [Corwin-P. ’15]. For all n1 ≥ · · · ≥ nk ≥ 1 and t ∈ Z≥0, E

• k
• i=1

qxni (t)+ni

• = (−1)kq

k(k−1) 2

(2πi)k

• γ1

· · ·

• γk
• A<B

zA − zB zA − qzB ×

k

• j=1

1 − zj 1 − νzj −nj1 + qJαzj 1 + αzj t dzj zj(1 − νzj) Proof By duality, the moments Et( n) = E k

i=1 qxni (t)+ni

• satisfy the

evolution equations Et+1( n) = ˜ Bα,qJαEt

• (

n) with the operator ˜ Bα,qJα in the right-hand side.

Asymptotics For the step initial data, one can write a Fredholm determinantal formula for the q-Laplace transform E

• 1
• ζqxn(t)+n; q
• ∞
• f the n-th particle location, and perform the asymptotic
• analysis. It is expected that

xn(κn) − cκn σκn1/3 → FGUE, n → ∞. This is known in particular cases: q-Hahn TASEP (α = −ν, qJα = −µ, 0 ≤ ν ≤ µ < 1)

[Veto ’14]

q-TASEP (q-Hahn TASEP with ν = 0)

[Ferrari-Veto ’13], [Barraquand ’14]

Finite I case When the number of vertical spins is bounded by I ∈ Z≥1 (e.g. for ASEP and the stochastic six-vertex model), the exclusion process should be defined differently. The solution (and Tracy-Widom asymptotics) can be obtained using a different approach [Tracy-Widom ’07], [Borodin–Corwin–Gorin ’14]: Write down transition probabilities Pt( x, y) — given by multiple contour integrals because they satisfy the evolution equations with right-hand side ˜ Bα,qJα. For special initial data (step or half-stationary), the

• bservable P(xn(t) ≤ x) can be computed with the help
• f certain symmetrization identities.

This observable is expressed as a Fredholm determinant, and thus one also gets Tracy-Widom asymptotics.

Symmetrization identities

General symmetrization identity The second Plancherel theorem applied to a specific function G( z) implies (after a change of variables)

• σ∈S(k)
• B<A

Sq,ν(ξσ(B), ξσ(A)) ξσ(A) − ξσ(B)

• n1≥...≥nk≥0
• i

(ν; q)ci (q; q)ci

k

• j=1

ξ

nj σ(j)

• =

1 − ν 1 − q k q(1 − ν) 1 − qν k(k−1)

2

k

• j=1

1 1 − ξj . Here ci are clusters of n (i.e., numbers of vertical arrows at each vertex), (a; q)m := (1 − a)(1 − aq) . . . (1 − aqm−1), and Sq,ν(ξ1, ξ2) := 1 − q 1 − qν + q − ν 1 − qν ξ2 + ν(1 − q) 1 − qν ξ1ξ2 − ξ1.

Symmetrization identity for the ASEP [Tracy-Widom ’07]

• σ∈S(k)
• B<A

Sq,ν(ξσ(B), ξσ(A)) ξσ(A) − ξσ(B)

• n1≥...≥nk≥0
• i

(ν; q)ci (q; q)ci

k

• j=1

ξnj

σ(j)

• =

1 − ν 1 − q k q(1 − ν) 1 − qν k(k−1)

2

k

• j=1

1 1 − ξj Setting q = 1/ν, the summation in n above can be done explicitly, and so one gets Tracy-Widom’s ASEP identity:

• σ∈S(k)

σ  

i<j

SASEP(ξi, ξj) ξj − ξi ξ2ξ2

3 . . . ξk−1 k

(1 − ξ1ξ2 . . . ξk)(1 − ξ2 . . . ξk) . . . (1 − ξk)   = q

k(k−1) 2

k

j=1(1 − ξj)

. where SASEP(ξ1, ξ2) := q − (1 + q)ξ1 + ξ1ξ2.

Thank you!