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

Dynamics on interlacing partitions for sl 2 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 = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram)

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states i λ i

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states i λ i ◮ Γ ± : half-vertex operators

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states ◮ Γ ± : half-vertex operators i λ i ◮ Γ + ( x ) : | λ � → � ν s ν / λ ( x )| ν �

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states i λ i ◮ Γ ± : half-vertex operators ◮ Γ + ( x ) : | λ � → � ν s ν / λ ( x )| ν �

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states ◮ Γ ± : half-vertex operators i λ i ◮ Γ + ( x ) : | λ � → � ν s ν / λ ( x )| ν � ◮ Γ − ( y ) : | λ � → � µ s λ / µ ( y )| µ �

We study statistics on ensembles of Young diagrams: why and how ? λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ 0 ) = (Young diagram) ◮ Motivating example: free fermions ◮ | λ � = : configuration of states i λ i ◮ Γ ± : half-vertex operators ◮ Γ + ( x ) : | λ � → � ν s ν / λ ( x )| ν � ◮ Γ − ( y ) : | λ � → � µ s λ / µ ( y )| µ �

◮ Matrix element s λ / µ ( z ) = � λ | Γ − ( z )| µ � = � µ | Γ + ( z )| λ � are the Schur functions ◮ For a diagonal observable O : | λ � → d λ | λ � � �O� + , − µ,λ ∝ � µ | Γ + ( x ) O Γ − ( y )| λ � = d ν s ν / µ ( x ) s ν / λ ( y ) ν ◮ Given partitions (states) µ,λ : � � 1 Schur measure S µ,λ ( ν ) = s ν / µ ( x ) s ν / λ ( y ) [Okounkov’01] Z µ,λ ◮ Expectations in the free fermions system and Schur measure are equivalent �O� + , − µ,λ = � d ν � S µ,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 s λ / µ � � s ν / µ ( x ) s ν / λ ( y ) = Π ( x ; y ) s µ / κ ( x ) s λ / κ ( y ) (Cauchy Identity) ν ν

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 s λ / µ � � s ν / µ ( x ) s ν / λ ( y ) = Π ( x ; y ) s µ / κ ( x ) s λ / κ ( y ) (Cauchy Identity) ν ν ν λ S µ,λ ( ν ) is interpreted as a transition probability from a state κ S µ,λ ( ν ) to a state ν . µ κ

� ◮ We build a random field of partitions � λ = { λ ( i , j ) } using local moves � � � � � (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 , 2 ) � 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 , 2 ) ◮ The average of observables in the � λ ( 1 , 1 ) λ ( 2 , 1 ) Schur random field corresponds to the expectation of observables in the � � � � free fermions system (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) 0 5 10 15 20 25 30 (Schur random field)           • • • • • • • • • • • • • • • • • • • • • • • • •  • • • • •   • • • • •   • • • • •   • • • • •   • • • • •            • • • • • • • • • • • • • • • • • • • • • • • • •                     • • • • • • • • • • • • • • • • • • • • • • • • •           • • • • • • • • • • • • • • • • • • • • • • • • •

◮ 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. (Schur Process)

◮ Consider the Schur process [Okounkov-Reshetikhin’03] λ ( 1 , t ) ≺ λ ( 2 , t ) ≺ · · · ≺ λ ( x , t ) pushTASEP TASEP ◮ A surprising fact is that the leftmost and rightmost diagonal evolve as autonomous Markov processes. (Schur Process) ◮ Coordinates on the leftmost diagonal rate 1 sample the TASEP (TASEP) ◮ Coordinates on the rightmost rate 1 diagonal sample a pushTASEP (pushTASEP) push

TRAIN OF THOUGHT Free fermions Cauchy Identities for special functions: � ν s ν / µ ( x ) s ν / λ ( y ) = Π ( x ; y ) � ν s µ / κ ( x ) s λ / 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 s ν / µ ( x ) s ν / λ ( y ) = Π ( x ; y ) � ν s µ / κ ( x ) s λ / 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 s λ / µ functions with the Macdonald functions P λ / µ , Q λ / µ : � � P ν / µ ( x ) Q ν / λ ( y ) = Π ( x ; y ) P µ / κ ( x ) Q λ / κ ( y ) ν ν we obtain the MacDonald Processes [Borodin-Corwin’11] (MacDonald Process)

Replacing the Schur s λ / µ functions with the Macdonald functions P λ / µ , Q λ / µ : � � P ν / µ ( x ) Q ν / λ ( y ) = Π ( x ; y ) P µ / κ ( x ) Q λ / κ ( y ) ν ν 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?

sl 2 Stochastic Vertex Models: ◮ Higher Spin Stochastic Six Vertex Model [Corwin-Petrov’15] : higher spin generalization of the symmetric ferroelectric ( ∆ > 1 ) six vertex model ◮ Probability of configuration of red path = product vertex weights t i 2 g g − 1 g + 1 g . . j 1 j 2 . g g g g i 1 2 q g + uθ 1 − q g uθ + q g us 1 − q g us L u,θ ( i 1 , j 1 ; i 2 , j 2 ) 1+ uθ 1+ uθ 1+ uθ 1+ uθ 1 0 Vertex weights depend on many x 1 2 3 · · · parameters ( q , s , u t ,θ x ) .

◮ 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 F λ / µ : spin Hall-Littlewood functions [Borodin’17] b F λ / µ : spin q -Whittaker functions [Borodin-Wheeler’17] (F / F Process)