= ( 1 2 0 ) = (Young diagram) We study statistics on - - PowerPoint PPT Presentation

Dynamics on interlacing partitions for sl2 stochastic vertex models

MUCCICONI MATTEO

based on collaborations with A. BUFETOV and L. PETROV

Colloquium in Physics

令和1年10月24日

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

◮ Γ±: half-vertex operators

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

◮ Γ±: half-vertex operators ◮ Γ+(x) : |λ →

ν sν/λ(x)|ν

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

◮ Γ±: half-vertex operators ◮ Γ+(x) : |λ →

ν sν/λ(x)|ν

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

◮ Γ±: half-vertex operators ◮ Γ+(x) : |λ →

ν sν/λ(x)|ν

◮ Γ−(y) : |λ →

µ sλ/µ(y)|µ

We study statistics on ensembles of Young diagrams: why and how ?

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) =

(Young diagram)

◮ Motivating example: free fermions

i λi

◮ |λ =

: configuration of states

◮ Γ±: half-vertex operators ◮ Γ+(x) : |λ →

ν sν/λ(x)|ν

◮ Γ−(y) : |λ →

µ sλ/µ(y)|µ

◮ Matrix element sλ/µ(z) = λ|Γ−(z)|µ = µ|Γ+(z)|λ

are the Schur functions

◮ For a diagonal observable O : |λ → dλ|λ O+,−

µ,λ ∝ µ|Γ+(x) O Γ−(y)|λ =

• ν

dν sν/µ(x) sν/λ(y) ◮ Given partitions (states) µ,λ:

Sµ,λ(ν) =

1 Zµ,λ sν/µ(x) sν/λ(y)

• Schur measure

[Okounkov’01]

• ◮ Expectations in the free fermions system and Schur measure are

equivalent

O+,−

µ,λ = dνSµ,λ

Γ+(x)Γ−(y) = Π(x; y)Γ−(y)Γ+(x), Π(x; y) ∈ C

Commutation relation between Γ− and Γ+ imply algebraic identities between functions sλ/µ

• ν

sν/µ(x) sν/λ(y) = Π(x; y)

• ν

sµ/κ(x) sλ/κ(y)

(Cauchy Identity)

Γ+(x)Γ−(y) = Π(x; y)Γ−(y)Γ+(x), Π(x; y) ∈ C

Commutation relation between Γ− and Γ+ imply algebraic identities between functions sλ/µ

• ν

sν/µ(x) sν/λ(y) = Π(x; y)

• ν

sµ/κ(x) sλ/κ(y)

(Cauchy Identity) κ µ λ ν

Sµ,λ(ν)

Sµ,λ(ν) is interpreted as a transition probability from a state κ to a state ν.

• (Schur random field)

◮ We build a random field of partitions λ = {λ(i,j)} using local moves

• λ(1,1)

(Schur random field)

◮ We build a random field of partitions λ = {λ(i,j)} using local moves

• λ(1,1)

λ(2,1)

(Schur random field)

◮ We build a random field of partitions λ = {λ(i,j)} using local moves

• λ(1,1)

λ(2,1) λ(1,2)

(Schur random field)

◮ We build a random field of partitions λ = {λ(i,j)} using local moves

• λ(1,1)

λ(2,1) λ(1,2)

(Schur random field)

◮ We build a random field of partitions λ = {λ(i,j)} using local moves ◮ The average of observables in the

Schur random field corresponds to the expectation of observables in the free fermions system

5 10 15 20 25 30

(Schur random field)

◮ Partitions on the same row (or

column) interlace

λ(1,t) ≺ λ(2,t) ≺ · · · ≺ λ(x,t) ◮ probability measures on interlacing

arrays appear in random matrix theory (eigenvalues of minors of self adjoint matrices)

     

• 

          

• 

          

• 

          

• 

          

• 

    

(Schur Process)

◮ Consider the Schur process

[Okounkov-Reshetikhin’03]

λ(1,t) ≺ λ(2,t) ≺ · · · ≺ λ(x,t)

(Schur Process)

◮ Consider the Schur process

[Okounkov-Reshetikhin’03]

λ(1,t) ≺ λ(2,t) ≺ · · · ≺ λ(x,t) ◮ A surprising fact is that the leftmost

and rightmost diagonal evolve as autonomous Markov processes.

TASEP pushTASEP (Schur Process)

rate 1

(TASEP)

rate 1 push

(pushTASEP)

◮ Consider the Schur process

[Okounkov-Reshetikhin’03]

λ(1,t) ≺ λ(2,t) ≺ · · · ≺ λ(x,t) ◮ A surprising fact is that the leftmost

and rightmost diagonal evolve as autonomous Markov processes.

◮ Coordinates on the leftmost diagonal

sample the TASEP

◮ Coordinates on the rightmost

diagonal sample a pushTASEP

TRAIN OF THOUGHT Free fermions Cauchy Identities for special functions:

• ν sν/µ(x) sν/λ(y) = Π(x; y)

ν sµ/κ(x) sλ/κ(y)

Random sampling of partitions Marginal processes of the field of random partitions might be interesting (TASEP, pushTASEP,etc.)

TRAIN OF THOUGHT Free fermions Cauchy Identities for special functions:

• ν sν/µ(x) sν/λ(y) = Π(x; y)

ν sµ/κ(x) sλ/κ(y)

Random sampling of partitions Marginal processes of the field of random partitions might be interesting (TASEP, pushTASEP,etc.) What processes arise when we consider Cauchy Identities for different special functions?

Replacing the Schur sλ/µ functions with the Macdonald functions

Pλ/µ,Qλ/µ:

• ν

Pν/µ(x) Qν/λ(y) = Π(x; y)

• ν

Pµ/κ(x) Qλ/κ(y)

we obtain the MacDonald Processes [Borodin-Corwin’11] (MacDonald Process)

Replacing the Schur sλ/µ functions with the Macdonald functions

Pλ/µ,Qλ/µ:

• ν

Pν/µ(x) Qν/λ(y) = Π(x; y)

• ν

Pµ/κ(x) Qλ/κ(y)

we obtain the MacDonald Processes [Borodin-Corwin’11] q-TASEP q-pushTASEP (MacDonald Process) Diagonals of the process are still markovian.

◮ q-TASEP and q-pushTASEP can be thought are discretizations of

the KPZ equation (stochastic PDE for growth of surfaces with lateral growth and relaxation)

◮ Algebraic properties (symmetries, operators, etc.) of Macdonald

functions allow an exact study of the marginal processes and of the KPZ equation.

QUESTION:

◮ What are the most general models that can be studied following

the MacDonald Processes scheme?

sl2 Stochastic Vertex Models: ◮ Higher Spin Stochastic Six Vertex Model [Corwin-Petrov’15] :

higher spin generalization of the symmetric ferroelectric (∆ > 1) six vertex model

x t 1 1 2 3 2 · · · . . .

◮ Probability of configuration of red

path = product vertex weights

j2 i2 j1 i1 g g g − 1 g g g + 1 g g

Lu,θ(i1, j1; i2, j2)

qg+uθ 1+uθ 1−qg 1+uθ uθ+qgus 1+uθ 1−qgus 1+uθ

Vertex weights depend on many parameters (q,s,ut,θx).

◮ The Stochastic Higher Spin Six Vertex Model generalizes a

number of stochastic integrable processes in the KPZ universality class: Asymmetric Exclusion Processes, Random Polymer Models, Random Walkers in Random Environmemnt, KPZ equation, etc.

◮ Does it admit a constructions as a marginal a field of interlacing

partitions?

RESULTS [Bufetov-M-Petrov’19,M-Petrov’??]

◮ We build a random field of partitions using

a Fλ/µ : spin Hall-Littlewood functions [Borodin’17] b Fλ/µ : spin q-Whittaker functions [Borodin-Wheeler’17]

(F/F Process)

RESULTS [Bufetov-M-Petrov’19,M-Petrov’??]

◮ We build a random field of partitions using

a Fλ/µ : spin Hall-Littlewood functions [Borodin’17] b Fλ/µ : spin q-Whittaker functions [Borodin-Wheeler’17]

HS6VM pushHS6VM (F/F Process) Technical points that we address:

◮ proof that diagonals are

autonomous Markov processes

◮ initiate theory of operators for

functions F,F

◮ give exact expressions to average

• f observables of the process

Summary of the talk

• 1. Free fermions and partitions: the Schur measure
• 2. Probability on interlacing partitions: Schur processes and random

symmetric matrices

• 3. Non free fermionic models: MacDonald processes
• 4. Taking the scheme to a more general level: the Higher Spin Six

Vertex Model OPEN QUESTIONS

◮ develop more the theory of operators of F,F functions ◮ study of the full F/F process ◮ connect the F/F process with the theory of Random Walkers in

Random environment