Stochastic vertex odels and syetric functions Alexey Bufetov MIT 6Noveber,2017
Applications of the algebra of syetric functions to probability: Schur easures (Okounkov), Schur processes (Okounkov-Reshetikhin), Macdonald processes (Borodin-Corwin). Algebra of syetric functions. Schur and Hall-Littlewood easures. Stochastic six vertex odel. Classical RSK algorith. Hall-Littlewood RSK algorith. Motivation of presented results: A better understanding of the structure of odels. Possible tools for the asyptotic analysis.
Consider a atrix {r ij } 1 i M,1 j N and define the quantity G(M, N) = ax P: up-right path (1, 1) (M, N) r ij. (i,j) P Exaple: G(2, 2) = 4, G(3, 2) = 8, G(2, 3) = 6, G(3, 3) = 9. 3 1 1 0 1 4 2 1 0
Assue that r i,j are i.i.d. rando variables with geoetric distribution P(r 1,1 = k) = (1 q)q k, 0 < q < 1, k = 0, 1, 2,... Johansson 99: N, N) a( )N li P G( s = F N b( )N 13 TW (s), where F TW (s) is a probability distribution function of Tracy-Wido distribution;, a( ), b( ) R. KPZ universality class. Siilar result if r i,j have Bernoulli distribution, but for strict up-right paths.
Young diagras: finite non-increasing sequences of integers = 1 2 3 4 0 1 2 3 4 5 6 1 2 3 4 1 = 6, 2 = 3, 3 = 2, 4 = 1 1 = 4, 2 = 3, 3 = 2,... = 1 + 2 + = 12, Y thesetofallyoungdiagras.
{x i } i=1 foralvariables. Newton power sus: Algebra of syetric functions p k = i=1 x k i. The algebra of syetric functions = R[p 1, p 2,...]. = 1 2 N, i Z 0. The Schur polynoial is defined by s (x 1,...,x N ) = det i,j=1,...,n x j +N j i 1 i<j N (x i x j ),
For t [0; 1) ahall-littlewoodpolynoialisdefinedvia Q (x 1,...,x N ; t) = c,t S N x 1 (1) x 2 (2)...x k (k) i<j x (i) tx (j) x (i) x (j). P (x 1,...,x N ; t) = ĉ,t Q (x 1,...,x N ; t). for soe explicit constants c,t, ĉ,t. For t = 0 Hall-Littlewood polynoials turn into Schur polynoials: Q (x 1,...,x N ;0) = s (x 1,...,x N ) Using Q (x 1,...,x N, 0; t) = Q (x 1,...,x N ; t), one can define Q, s. {s } Y linearbasisin. {Q } Y linearbasisin.
Cauchy identity: 1 tx i y j P (x 1,...,x N ; t)q (y 1,...,y N ; t) = Y i,j 1 x i y j a i b j < 1, a i > 0, b j > 0. Schur easure on Young diagras: Prob( ) = (1 a i b j ) s (a 1,...,a M )s (b 1,...,b N ). i,j Hall-Littlewood easure: Prob( ) = i,j 1 a i b j 1 ta i b j P (a 1,...,a M ; t)q (b 1,...,b N ; t).
Let r ij be independent rando variables with geoetric distribution Prob(r ij = x) = (1 a i b j )(a i b j ) x, x = 0, 1, 2,... G(M, N) = ax P: up-right path (1, 1) (M, N) r ij. (i,j) P Then G(M, N) has the sae distribution as the length of the first row of the (rando) Young diagra distributed according to the Schur easure with paraeters a 1,...,a M, b 1,...,b N. This is a key fact in the analysis of the asyptotic behavior of G(M, N) (then one uses deterinantal processes and the steepest descent analysis). More generally, one can use syetric functions for analyzing ulti-point distribution: M 1 M k and N 1 N k {G(M i, N i )}.
Six vertex odels are of interest as odels of statistical echanics ( square ice ). H H H H H O H O H O H O H H H H H H O H O H O H O H H H H H H O H O H O H O H
Consider one particular odel: for 0 < t < 1, 0 p 2 < p 1 1 let the weights have the for 1 1 p 1 1 p 1 p 2 1 p 2 Boundary conditions: quadrant, all paths enter fro the left. This is a stochastic six vertex odel introduced by Gwa-Spohn 92, and recently studied in Borodin-Corwin-Gorin 14. It has a degeneration into ASEP (asyptotics of height function Tracy-Wido 07)
Height function: 4 3 2 1 1 3 2 1 1 0 2 1 1 0 0 1 1 0 0 0 0 0 0 0 0
Height function: b 4 b 3 b 2 b 1 4 3 2 1 1 3 2 1 1 0 2 1 1 0 0 1 1 0 0 0 0 0 0 0 0 a 1 a 2 a 3 a 4
1 1 1 a i b j 1 ta i b j (1 t)a i b j 1 ta i b j t(1 a i b j ) 1 ta i b j 1 t 1 ta i b j Borodin-Bufetov-Wheeler 16 the height function H(M, N) for astochasticsixvertexodelwithweightsaboveisdistributed as N 1 (M, N), where is distributed as Hall-Littlewood easure with paraeters a 1,...,a M, b 1,...,b N. Borodin-Bufetov-Wheeler 16 More generally, for M 1 M k and N 1 N k the height functions {H(M i, N i )} is distributed as first coluns of diagras fro Hall-Littlewood process.
How to see the full Young diagra? RSK algorith: RSK-algorith (Robinson,Schensted,Knuth). Foin s growth diagra: F Y Y Y Z Y. INPUT: three Young diagras µ, µ, r Z 0. OUTPUT: Young diagra such that = µ + r.,, also r r µ µ
Set (k, 0) =, (0, k) =, for any k Z 0. Then, define inductively (k + 1, l + 1) = F ( (k, l), (k, l + 1), (k + 1, l), r kl ). That is, we add boxes one by one using eleentary steps described before. Note that by construction for any (k, l) we have (k, l) (k + 1, l), (k, l) (k, l + 1). (0, 0) (1, 3) (2, 3) r 13 r 23 (1, 2) (2, 2) (3, 2) (4, 2) r 12 r 22 r 32 r 42 (1, 1) (2, 1) (3, 1) (4, 1) r 11 r 21 r 31 r 41
Applications to Schur easures and Schur processes Let r ij be independent rando variables with geoetric distribution Prob(r ij = x) = (1 a i b j )(a i b j ) x, x = 0, 1, 2,... Then (M, N) is distributed according to the Schur easure with paraeters a 1,...,a M, b 1,...,b N. More generally, for M 1 M k and N 1 N k the faily { (M i, N i )} is distributed as a Schur process. 1(M, N) coincides with G(M, N). RSK for Hall-Littlewood functions?
Properties of classical RSK: 1) Saples Schur easures and processes. 2) Markov projection for the first row / colun. 3) Syetry: F (µ,,, r) = F (µ,,, r). 4) Local interaction. And a lot of other structure... (jeu de taquin, plactic onoid, etc, etc). The generalization of these properties to Hall-Littlewood functions is interesting fro both probabilistic and cobinatorial points of view. (q-whittaker functions) Recent rando RSK-algoriths for generalizations of Schur functions: O Connell-Pei 12, Borodin-Petrov 13, Bufetov-Petrov 14, Matveev-Petrov 15.
INPUT: three Young diagras µ, µ, r Z 0. OUTPUT: Rando (!) Young diagra such that,, also = µ + r. r r µ µ Deterined by coe cients U r ( µ ).
a i, b j R >0, i, j N, a i b j < 1 (1, 3) (2, 3) b 3 b 2 b 1 r 13 r 23 (1, 2) (2, 2) (3, 2) (4, 2) r 12 r 22 r 32 r 42 (1, 1) (2, 1) (3, 1) (4, 1) r 11 r 21 r 31 r 41 (0, 0) a 1 a 2 a 3 a 4 P(r i,j = d) = (1 t1 d 1 )(a i b j ) d 1 a ib j 1 ta i b j, d = 0, 1, 2,... We have P( (, n) = ) P (a 1,...,a M )Q (b 1,...,b N ).
Bufetov-Matveev 17: Hall-Littlewood RSK field. Properties. Saples Hall-Littlewood easures and processes analogously to the Schur case. The distribution of the first colun gives the height function in the stochastic six vertex odel. Cobinatorial structure naturally generalize the Schur case. Bufetov-Matveev 17: based on Borodin-Corwin-Gorin-Shakirov 13 forulas for Hall-Littlewood processes. Thus, the Hall-Littlewood RSK field is an integrable object.
There is a liit fro the stochastic six vertex odel to ASEP. Bufetov-Matveev 17 2-layer ASEP (also integrable). v = 1 v = 1 t v = 1 t 1 t k v = t tk 1 t k v = t v = t v = t v = t v = t v = 1 v = 1 v = 1 v = 1 v = t
1 + 2 + 1 1 1 a i b j 1 t +1 a 1 ta i b j 1 t a + 1 + 1 + 1 t(1 a i b j ) 1 ta i b j + 1 b t b t +1 + 1 + 1 + 1 + 1 + 1 Bufetov-Petrov 17+,in progress the height function H(M, N) for a (dynaical) stochastic six vertex odel with weights above is distributed as 1 (M, N), where is distributed as the spin Hall-Littlewood easure with paraeters a 1,...,a M, b 1,...,b N. Bufetov-Petrov 17+,in progress More generally, for M 1 M k and N 1 N k the height functions {H(M i, N i )} is distributed as first coluns of diagras fro the spin Hall-Littlewood process.
Equality in distribution between stochastic six vertex odel and Hall-Littlewood easure/process. Cobinatorics: RSK algorith provides an extension of this result. Further directions: vertex odels / Yang-Baxter equation /quantugroupsvs.syetricfunctions/algebraic cobinatorics. Asyptotics of odels. A. Borodin, A. Bufetov, M. Wheeler, Between the stochastic six vertex odel and Hall-Littlewood processes, arxiv:1611.09486. A. Bufetov, K. Matveev, Hall-Littlewood RSK field, arxiv:1705.07169.