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.