Yang Baxter Field and StochasticVertex Models
MUCCICONI MATTEO
based on a collaboration with A. BUFETOV and L. PETROVStochastic Analysis, Random Fields and Integrable
Probability

令和1年08月5日

Yang Baxter Field and Stochastic Vertex Models
IWe discuss a random algorithm whose output is a random"field"of Young Diagrams λ = {λ(x, y ) | x, y ∈ Z ≥0 }.
yλ(x, y ) =x

Yang Baxter Field and Stochastic Vertex Models
IWe discuss a random algorithm whose output is a random"field"of Young Diagrams λ = {λ(x, y ) | x, y ∈ Z ≥0 }.
yλ(x, y ) =x
IThe algorithm consists in a sequence of local moves defined
through a "bijectivization"of the Yang-Baxter Equation=+w(a1 )w(a2 )+w(b1 )w(b2 )

Yang Baxter Field and Stochastic Vertex Models
IWe discuss a random algorithm whose output is a random"field"of Young Diagrams λ = {λ(x, y ) | x, y ∈ Z ≥0 }.
yλ(x, y ) =xIThe algorithm consists in a sequence of local moves defined
through a "bijectivization"of the Yang-Baxter Equation=+w(a1 )Iw(a2 )+w(b1 )w(b2 )A bijectivization [Bufetov-Petrov ’17] is a way to assign
transition probabilities
P(ai → bj ).

Yang Baxter Field and Stochastic Vertex Models
I

Random fields of Young diagrams are popular objects in
Integrable Probability because they allow the description ofrandom tilings, stochastic interacting particle systems, stochastic
vertex models, etc.

Yang Baxter Field and Stochastic Vertex Models
I

Random fields of Young diagrams are popular objects in
Integrable Probability because they allow the description ofrandom tilings, stochastic interacting particle systems, stochastic
vertex models, etc.

IIn our case, the random field inherits its integrability from theYang-Baxter Equation. The joint probability distribution along
down-right paths π assume the form
F˙1≺y
τ (0,4)u4τ (0,3) λ(1,3) λ(2,3) λ(3,3) λ(4,3)τ (0,2) λ(1,2) λ(2,2) λ(3,2) λ(4,2)
u2

≺1u3

P({λ(x, y ) }(x, y )∈π ) = Z1π

Îi : yi +1 =yi −1 Fλ(xi , yi ) /λ(xi +1 , yi +1 ) (uyi )τ (0,1) λ(1,1) λ(2,1) λ(3,1) λ(4,1)
u1xτ (0,0) τ (1,0) τ (2,0) τ (3,0) τ (4,0) τ (5,0)
v1v2v3v4v5G×Î

i : xi +1 =xi +1 Gλ(xi +1 , yi +1 ) /λ(xi , yi ) (vxi +1 )where F, G are special symmetric functions (multi-parameter
generalizations of the Schur polynomials).

Yang Baxter Field and Stochastic Vertex Models
Results we obtain considering this simple procedure:
IFrom marginal observables (e.g. λ(x, y ) → length(λ(x, y ) )) werecover the Stochastic Six Vertex Model [Gwa-Spohn ’92], theStochastic Higher Spin Six Vertex Model [Corwin-Petrov ’15]
and also more general vertex models

Yang Baxter Field and Stochastic Vertex Models
Results we obtain considering this simple procedure:II

From marginal observables (e.g. λ(x, y ) → length(λ(x, y ) )) werecover the Stochastic Six Vertex Model [Gwa-Spohn ’92], theStochastic Higher Spin Six Vertex Model [Corwin-Petrov ’15]
and also more general vertex models
We are able to construct "MacDonald operators"for symmetricfunctions F, G. These are guessed from known results aboutstochastic vertex models

Yang Baxter Field and Stochastic Vertex Models
Results we obtain considering this simple procedure:I

From marginal observables (e.g. λ(x, y ) → length(λ(x, y ) )) werecover the Stochastic Six Vertex Model [Gwa-Spohn ’92], theStochastic Higher Spin Six Vertex Model [Corwin-Petrov ’15]
and also more general vertex models

I

We are able to construct "MacDonald operators"for symmetricfunctions F, G. These are guessed from known results aboutstochastic vertex models

IWe describe statistics of the random field using these newoperators (like in MacDonald Processes [Borodin-Corwin ’11])