Stochastic vertex models and symmetric

functions

Alexey Bufetov

MIT

6 November, 2017

Applications of the algebra of symmetric functions toprobability: Schur measures (Okounkov), Schur processes(Okounkov-Reshetikhin), Macdonald processes(Borodin-Corwin).

Algebra of symmetric functions. Schur andHall-Littlewood measures.

Stochastic six vertex model.

Classical RSK algorithm.

Hall-Littlewood RSK algorithm.

Motivation of presented results: A better understanding of thestructure of models. Possible tools for the asymptotic analysis.

Consider a matrix {rij

}1≤i≤M,1≤j≤N and define the quantity

G(M ,N) = maxP: up-right path (1,1)→ (M ,N) �(i ,j)∈P rij .

Example: G(2,2) = 4, G(3,2) = 8, G(2,3) = 6, G(3,3) = 9.

2 1 0

0 1 4

3 1 1

Assume that ri ,j are i.i.d. random variables with geometric

distribution P(r1,1 = k) = (1 − q)qk , 0 < q < 1, k = 0,1,2, . . . .Johansson’99:

limN→∞P �G(��N�,N) − a(�)N

b(�)N1�3 ≤ s� = FTW

(s),where F

TW

(s) is a probability distribution function ofTracy-Widom distribution; �, a(�),b(�) ∈ R.KPZ universality class.

Similar result if ri ,j have Bernoulli distribution, but for strict

up-right paths.

Young diagrams: finite non-increasing sequences of integers� = �1 ≥ �2 ≥ �3 ≥ �4 ≥ ⋅ ⋅ ⋅ ≥ 0

�1

�2

�3

�4

�′1 �′2 �′3 �′4 �′5 �′6

�1 = 6,�2 = 3,�3 = 2,�4 = 1�′1 = 4,�′2 = 3,�′3 = 2, . . . .��� ∶= �1 + �2 + ⋅ ⋅ ⋅ = 12, Y — the set of all Young diagrams.

Algebra of symmetric functions

{xi

}∞i=1 — formal variables.

Newton power sums:

pk

∶= ∞�i=1

xki

.

The algebra of symmetric functions ⇤ ∶= R[p1,p2, . . . ].� = �1 ≥ �2 ≥ ⋅ ⋅ ⋅ ≥ �N

, �i

∈ Z≥0.The Schur polynomial is defined by

s�(x1, . . . , xN) ∶= deti ,j=1,...,N �x�j

+N−ji

�∏1≤i<j≤N(xi − xj) ,

For t ∈ [0; 1) a Hall-Littlewood polynomial is defined via

Q�(x1, . . . , xN ; 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�(x1, . . . , xN ; t) ∶= c�,tQ�(x1, . . . , xN ; t).for some explicit constants c�,t , c�,t .For t = 0 Hall-Littlewood polynomials turn into Schurpolynomials:

Q�(x1, . . . , xN ; 0) = s�(x1, . . . , xN)Using Q�(x1, . . . , xN ,0; t) = Q�(x1, . . . , xN ; t), one can defineQ� ∈ ⇤, s� ∈ ⇤.{s�}�∈Y — linear basis in ⇤. {Q�}�∈Y — linear basis in ⇤.

Cauchy identity:

��∈Y

P�(x1, . . . , xN ; t)Q�(y1, . . . , yN ; t) =�i ,j

1 − txi

yj

1 − xi

yj

ai

bj

< 1, ai

> 0,bj

> 0. Schur measure on Young diagrams:

Prob(�) =�i ,j

(1 − ai

bj

) s�(a1, . . . , aM)s�(b1, . . . ,bN).Hall-Littlewood measure:

Prob(�) =�i ,j

1 − ai

bj

1 − tai

bj

P�(a1, . . . , aM ; t)Q�(b1, . . . ,bN ; t).

Let rij

be independent random variables with geometricdistribution Prob(r

ij

= x) = (1 − ai

bj

)(ai

bj

)x , x = 0,1,2, . . . .G(M ,N) = max

P: up-right path (1,1)→ (M ,N) �(i ,j)∈P rij .Then G(M ,N) has the same distribution as the length of thefirst row of the (random) Young diagram distributed accordingto the Schur measure with parameters a1, . . . , aM ,b1, . . . ,bN .

This is a key fact in the analysis of the asymptotic behavior ofG(M ,N) (then one uses determinantal processes and thesteepest descent analysis).

More generally, one can use symmetric functions for analyzingmulti-point distribution: M1 ≥ ⋅ ⋅ ⋅ ≥Mk

and N1 ≤ ⋅ ⋅ ⋅ ≤ Nk{G(Mi

,Ni

)}.

Six vertex models are of interest as models of statisticalmechanics (“square ice”).

O O O O

O O O O

O O O O

H H H H H

H H H H H

H H H H H

H H H H

H H H H

H H H H

Consider one particular model: for 0 < t < 1, 0 ≤ p2 < p1 ≤ 1 letthe weights have the form

1 1 p1 1 − p1 p2 1 − p2

Boundary conditions: quadrant, all paths enter from the left.

This is a stochastic six vertex model introduced byGwa-Spohn’92, and recently studied inBorodin-Corwin-Gorin’14.

It has a degeneration into ASEP (asymptotics of heightfunction Tracy-Widom’07)

Height function:

0 0 0 0 0

1 1 0 0 0

2 1 1 0 0

3 2 1 1 0

4 3 2 1 1

Height function:

0 0 0 0 0

1 1 0 0 0

2 1 1 0 0

3 2 1 1 0

4 3 2 1 1

a1 a2 a3 a4

b1

b2

b3

b4

1 11−a

i

b

j

1−tai

b

j

(1−t)ai

b

j

1−tai

b

j

t(1−ai

b

j

)1−ta

i

b

j

1−t1−ta

i

b

j

Borodin-Bufetov-Wheeler’16 the height function H(M ,N) fora stochastic six vertex model with weights above is distributedas N − �′1(M ,N), where � is distributed as Hall-Littlewoodmeasure with parameters a1, . . . , aM , b1, . . . ,bN .

Borodin-Bufetov-Wheeler’16 More generally, for M1 ≥ ⋅ ⋅ ⋅ ≥Mk

and N1 ≤ ⋅ ⋅ ⋅ ≤ Nk

the height functions {H(Mi

,Ni

)} isdistributed as first columns of diagrams from Hall-Littlewoodprocess.

How to see the full Young diagram ?

RSK algorithm: RSK-algorithm (Robinson,Schensted,Knuth).Fomin’s growth diagram: F ∶ Y ×Y ×Y ×Z→ Y.INPUT: three Young diagrams µ � �, µ � ⌫, r ∈ Z≥0.OUTPUT: Young diagram ⇢ such that � � ⇢, ⌫ � ⇢, also�⇢� − ��� = �⌫� − �µ� + r .

rµ

�

⌫ µ

�

⌫

⇢

r

Set �(k ,0) = �, �(0, k) = �, for any k ∈ Z≥0. Then, defineinductively

�(k + 1, l + 1) = F (�(k , l),�(k , l + 1),�(k + 1, l), rkl

).That is, we add boxes one by one using elementary stepsdescribed before.Note that by construction for any (k , l) we have�(k , l) � �(k + 1, l), �(k , l) � �(k , l + 1).

�

�

(0,0)r11 r21 r31 r41

r12 r22 r32 r42

r13 r23

�(1,1) �(2,1) �(3,1) �(4,1)

�(1,2) �(2,2) �(3,2) �(4,2)

�(1,3) �(2,3)

Applications to Schur measures and Schur processes

Let rij

be independent random variables with geometricdistribution Prob(r

ij

= x) = (1 − ai

bj

)(ai

bj

)x , x = 0,1,2, . . . .Then �(M ,N) is distributed according to the Schur measurewith parameters a1, . . . , aM ,b1, . . . ,bN .

More generally, for M1 ≥ ⋅ ⋅ ⋅ ≥Mk

and N1 ≤ ⋅ ⋅ ⋅ ≤ Nk

the family{⇤(Mi

,Ni

)} is distributed as a Schur process.

�1(M ,N) coincides with G(M ,N).RSK for Hall-Littlewood functions ?

Properties of classical RSK:

1) Samples Schur measures and processes.2) “Markov projection” for the first row / column.3) Symmetry: F (µ,�, ⌫, r) = F (µ, ⌫,�, r).4) Local interaction.And a lot of other structure... (jeu de taquin, plactic monoid,etc, etc).

The generalization of these properties to Hall-Littlewoodfunctions is interesting from both probabilistic andcombinatorial points of view.

(q-Whittaker functions) Recent random RSK-algorithms forgeneralizations of Schur functions: O’Connell-Pei’12,Borodin-Petrov’13, Bufetov-Petrov’14, Matveev-Petrov’15.

INPUT: three Young diagrams µ � �, µ � ⌫, r ∈ Z≥0.OUTPUT: Random (!) Young diagram ⇢ such that � � ⇢,⌫ � ⇢, also �⇢� − ��� = �⌫� − �µ� + r .

rµ

�

⌫ µ

�

⌫

⇢

r

Determined by coe�cients U r(�→ ⇢ � µ→ ⌫).

ai

,bj

∈ R>0, i , j ∈ N, ai

bj

< 1

�

�

(0,0)r11 r21 r31 r41

r12 r22 r32 r42

r13 r23

�(1,1) �(2,1) �(3,1) �(4,1)

�(1,2) �(2,2) �(3,2) �(4,2)

�(1,3) �(2,3)

a1 a2 a3 a4

b1

b2

b3

P(ri ,j = d) = (1 − t1d≥1)(aibj)d 1 − a

i

bj

1 − tai

bj

, d = 0,1,2, . . .We have P(�(m,n) = �) ∼ P�(a1, . . . , aM)Q�(b1, . . . ,bN).

Bufetov-Matveev’17: Hall-Littlewood RSK field. Properties.

Samples Hall-Littlewood measures and processesanalogously to the Schur case.

The distribution of the first column gives the heightfunction in the stochastic six vertex model.

Combinatorial structure naturally generalize the Schurcase.

Bufetov-Matveev’17: based onBorodin-Corwin-Gorin-Shakirov’13 — formulas forHall-Littlewood processes.

Thus, the Hall-Littlewood RSK field is an integrable object.

There is a limit from the stochastic six vertex model to ASEP.

Bufetov-Matveev’17 2-layer ASEP (also integrable).

�v = 1

�v = 1

�v = 1

�v = 1

�v = t

�v = t

�v = t

�v = t

�v = t

�v = t

�v = 1

�v = 1 − t

�v = 1−t

1−tk�

v = t−tk1−tk

1

m

m + 1m + 1m + 2

1

mm

mm

1−ai

b

j

1−tai

b

j

1−tm+1a1−tma

mm

m + 1m + 1mm

m

m + 1t(1−a

i

b

j

)1−ta

i

b

j

b−tmb−tm+1

m

m + 1m

m + 1m

m + 1m + 1m + 1

Bufetov-Petrov’17+,in progress the height function H(M ,N)for a (dynamical) stochastic six vertex model with weightsabove is distributed as �′1(M ,N), where � is distributed as thespin Hall-Littlewood measure with parameters a1, . . . , aM ,b1, . . . ,bN .Bufetov-Petrov’17+,in progress More generally, forM1 ≥ ⋅ ⋅ ⋅ ≥Mk

and N1 ≤ ⋅ ⋅ ⋅ ≤ Nk

the height functions{H(Mi

,Ni

)} is distributed as first columns of diagrams fromthe spin Hall-Littlewood process.

Equality in distribution between stochastic six vertexmodel and Hall-Littlewood measure/process.

Combinatorics: RSK algorithm provides an extension ofthis result.

Further directions: vertex models / Yang-Baxter equation/ quantum groups vs. symmetric functions / algebraiccombinatorics. Asymptotics of models.

A. Borodin, A. Bufetov, M. Wheeler, “Between thestochastic six vertex model and Hall-Littlewoodprocesses”, arXiv:1611.09486.

A. Bufetov, K. Matveev, “Hall-Littlewood RSK field”,arXiv:1705.07169.