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 − 1 g 1 + α q g α (1 − q g ) 0 0 0 1 1 + α 1 + α g g g + 1 g 1 − ν q g α + ν q g 1 0 1 1 1 + α 1 + α g g

Vertex weights ( q , ν = q − I , α, J = 1 ) g g − 1 1 + α q g α (1 − q g ) 0 0 0 1 1 + α 1 + α g g g + 1 g 1 − ν q g α + ν q g 1 0 1 1 1 + α 1 + α g g 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 orthogonal polynomials, or basic hypergeometric functions 4 φ 3 .

General higher spin vertex model ( q , ν, α ; J ∈ Z ≥ 1 ) — obtained by fusion from the J = 1 case. One can have q J ∈ C (because general J vertex weights are polynomial in q J ). The model is stochastic if, for example, J ∈ Z ≥ 1 ; or α = − ν , q J α = − 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 exclusion process / zero range process q -Hahn TASEP Discrete time q -TASEP Stochastic six-vertex model [Korepin, Bogoli- ubov, Izergin, ’97] (ABA) q -TASEP ASEP Strict/weak [Kirillov–Reshetikhin polymer ’87] (fusion) Brownian motions Semi-discrete Brownian polymer with skew reflection [Mangazeev ’14] (R matrices) KPZ equation / SHE / continuum polymer [Borodin ’14] [Corwin–P. ’15] KPZ fixed point (e.g. Tracy-Widom distributions)

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 ( n 1 ≥ n 2 ≥ . . . ≥ n k ). The transition operator (= transfer matrix) ˜ B α, q J α . z = ( z 1 , . . . , z k ) ∈ C k : Eigenfunctions depend on � k � ˜ 1 + q J α z j B α, q J α Ψ ℓ � Ψ ℓ � ( � n ) = z ( � n ) z � � 1 + α z j j =1

Left eigenfunctions k � ˜ 1 + q J α z j B α, q J α Ψ ℓ � � Ψ ℓ ( � z ( � n ) = n ) , � � z 1 + α z j j =1 � � q j α + ν � 1 − z i (under condition � < 1 for all i and 1 ≤ j ≤ J − 1), � � 1 − ν z i 1+ q j α where � 1 − z σ ( j ) k � − n j z σ ( A ) − qz σ ( B ) � � � Ψ ℓ z ( � n ) = . � z σ ( A ) − z σ ( B ) 1 − ν z σ ( j ) σ ∈ S ( k ) 1 ≤ B < A ≤ k j =1 Remark B α, q J α depend on ( q , ν, α, J ), and eigenfunctions The operators ˜ — only on ( q , ν ): commuting system of transfer matrices. Remark The eigenfunctions are “algebraic”: they are not compactly sup- n (which is a natural domain for ˜ B α, q J α ). ported in � Cf. exponents as eigenfunctions of d / dx .

Left eigenfunctions k � ˜ 1 + q J α z j B α, q J α Ψ ℓ � � Ψ ℓ ( � z ( � n ) = n ) , � � z 1 + α z j j =1 where � 1 − z σ ( j ) k � − n j z σ ( A ) − qz σ ( B ) � � � Ψ ℓ z ( � n ) = . � z σ ( A ) − z σ ( B ) 1 − ν z σ ( j ) 1 ≤ B < A ≤ k j =1 σ ∈ S ( k ) Right eigenfunctions � 1 − z σ ( j ) k � n j qz σ ( A ) − z σ ( B ) � � � Ψ r z ( � n ) = C q ,ν ( � n ) , � z σ ( A ) − z σ ( B ) 1 − ν z σ ( j ) B < A j =1 σ ∈ S ( k ) with k 1 + q J α z j z ˜ B α, q J α � � Ψ r Ψ r � ( � n ) = z . � � 1 + α z j j =1 B α, q J α is not (Hermitian) symmetric, but is con- The operator ˜ jugate to its transpose ( PT-symmetry ).

Solving particle systems Goal Understand large-time behavior of the system, i.e., raise the B α, q J α to a large power, and preferably be matrix (operator) ˜ 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+] ). L R R x 1 x 2 x 3 x k 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 x 1 < x 2 < . . . < x k .

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

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

Coordinate Bethe ansatz for ASEP k = 1 : H (1) f ( x 1 ) = R( f ( x 1 + 1) − f ( x 1 )) + L( f ( x 1 − 1) − f ( x 1 )) . k = 2, x 1 + 1 < x 2 : H (2) f ( x 1 , x 2 ) = R( f ( x 1 + 1 , x 2 ) − f ( x 1 , x 2 )) + L( f ( x 1 − 1 , x 2 ) − f ( x 1 , x 2 )) + R( f ( x 1 , x 2 + 1) − f ( x 1 , x 2 )) + L( f ( x 1 , x 2 − 1) − f ( x 1 , x 2 )) H (1) + H (1) � � = f ( x 1 , x 2 ) . 1 2 k = 2, x 1 + 1 = x 2 : x 1 cannot jump right, x 2 cannot jump left H (2) f ( x 1 , x 2 ) = R( f ( x 1 , x 2 + 1) − f ( x 1 , x 2 )) + L( f ( x 1 − 1 , x 2 ) − f ( x 1 , x 2 )) H (1) + H (1) � � = f ( x 1 , x 2 ) + discrepancy , 1 2 discrepancy = R f ( x 1 + 1 , x 2 ) + L f ( x 1 , x 2 − 1) − f ( x 1 , x 2 )

Coordinate Bethe ansatz for ASEP When x 1 + 1 = x 2 , discrepancy = R f ( x 1 + 1 , x 2 ) + L f ( x 1 , x 2 − 1) − f ( x 1 , x 2 ) involves values of f outside the “physical region” x 1 < x 2 . 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 (1) + . . . + H (1) H ( k ) f = � � f if f such that for any i , 1 k R f ( . . . , x i +1 , x i +1 , . . . )+L f ( . . . , x i , x i +1 − 1 , . . . ) − f ( . . . ) = 0 whenever x i + 1 = x i +1 .

Coordinate Bethe ansatz for ASEP Therefore, one can diagonalize each H (1) separately, and i combine the eigenfunctions so that to satisfy the boundary conditions. The sum of one-particle operators has eigenfunctions � 1 + z σ ( i ) k � − x i � � z = ( z 1 , . . . , z k ) ∈ C k . A σ ( � z ) , � 1 + z σ ( i ) / q i =1 σ ∈ S ( k ) These will be eigenfunctions for any choice of A σ ( � z ). Then it is possible to choose A σ ( � z ) to satisfy the boundary conditions, and thus one has ASEP eigenfunctions � 1 + z σ ( i ) k � − x i z σ ( B ) − qz σ ( A ) � � � z σ ( B ) − z σ ( A ) 1 + z σ ( i ) / q σ ∈ S ( k ) B < A i =1

Coordinate Bethe ansatz for the higher spin vertex model