Connecting q-Whittaker and periodic Schurmeasures

T. Sasamoto

(Joint work with T. Imamura, M. Mucciconi)

Aug 2021 @ Stanford seminar

arXiv: 2106.11913, arXiv: 2106.11922, arXiv: 2108(?).xxxxx

1

Summary• Identity bet. marginals of q-Whittaker & periodic Schur meas.

• PNG and TASEP are related to Schur measure, determinantal

point process (DPP) of free fermion at zero temperature.

• Most KPZ models are related to q-Whittaker measure. By

standard approach, one arrives at Fredholm determinant

formula after long calculations using Bethe ansatz.

Asymptotics are rather involved. Half space is difficult.

• Our identity provides a new approach to study KPZ models

through DPP of free fermions at finite temperature.

Asymptotics is standard. Half space models can be studied.

• A proof is bijective through a generalization of RSK, which

shows similar behaviors to box-ball systems.

2

Plan

1. Our new identity connecting the two measures

2. PNG, TASEP and Schur measure

Last passage percolation

3. KPZ equation and T > 0 polymer

Fredholm det with T > 0 kernel

4. Discrete KPZ models and q-Whittaker measure

Periodic Schur measure

5. Bijection

6. A new approach to KPZ models

3

1. IntroductionCauchy identities for three polynomials

a = (a1, . . . , aN), b = (b1, . . . , bM), P : set of partitions

Schur ∑λ∈P

sλ(a)sλ(b) =

N∏i=1

M∏j=1

1

1 − aibj

q-Whittaker∑µ∈P

Pµ(a)Qµ(b) =N∏i=1

M∏j=1

1

(aibj; q)∞(=: ZqW)

Skew Schur∑λ,ρ∈Pρ⊂λ

q|ρ|sλ/ρ(a)sλ/ρ(b) =1

(q; q)∞

N∏i=1

M∏j=1

1

(aibj; q)∞(=: ZpS)

4

Our new identity relating q-Whittaker and skew Schur

Combining the last two identities and writing1

(q; q)∞=

∑ν∈P

q|ν|

∑µ,ν∈P

q|ν|Pµ(a)Qµ(b) =∑

λ,ρ∈Pρ⊂λ

q|ρ|sλ/ρ(a)sλ/ρ(b)

Our new identity is the following refinement.

Theorem:∑µ,ν∈P

µ1+ν1≤n

q|ν|Pµ(a)Qµ(b) =∑

λ,ρ∈Pρ⊂λ,λ1≤n

q|ρ|sλ/ρ(a)sλ/ρ(b)

Two proofs: Matching of Fredholm determinants for both sides

and a bijective one.

Today we mainly explain the first one in connection to KPZ.

5

q-Whittaker and periodic Schur measuresq-Whittaker measure

MqW(µ) =1

ZqWPµ(a)Qµ(b)

Periodic Schur measure

MpS(λ) =1

ZpS

∑ρ∈P,ρ(⊂λ)

q|ρ|sλ/ρ(a)sλ/ρ(b)

Let us also introduce two independent random variables χ, S:

P(χ = n) =qn

(q; q)n(q; q)∞, n = 0, 1, 2, . . .

P(S = ℓ) =tℓqℓ

2/2

(q; q)∞θ(−tq1/2), ℓ ∈ Z, for t > 0

with θ(x) = (x; q)∞(q/x; q)∞

6

Rewriting of the identity

• The identity can be written as the one for marginals

P (µ1 + χ ≤ n) = P (λ1 ≤ n)

• Shift mixed periodic Schur measure is a DPP. Using

P(χ + S ≤ n) =1

(−tq12+n; q)∞

the identity can be written as

E

[1

(−tq12+n−µ1; q)∞

]= P(λ1 + S ≤ n)

Both hand sides can be written as Fredholm determinants

(RHS: DPP, LHS: Connection to KPZ).

7

Kardar-Parisi-Zhang (KPZ) universalityNonequilibrium statistical physics: fluctuations in surface growth.

Experiments by liquid crystal (2010 Takeuchi Sano)

Fluctuations: O(t1/3) and scaled distributions

8

2. PNG, TASEP and Schur measurePNG (polynuclear growth) model

0

20

40

60

80

100

120

140

160

180

200

-200 -150 -100 -50 0 50 100 150 200

"PNG_st.dat"

t=0

t=1

t=2

t=3

t=4

xy

t

r

1

2

4

5

6

8

3

7

T

01 2 3 4 5 6 7 8

6 4 7 1 3 8 2 5

1 2 53 7 846

1 3 62 5 847

P Q Standard Taubleaux9

1. PNG, TASEP and Schur measurePNG (polynuclear growth) model

0

20

40

60

80

100

120

140

160

180

200

-200 -150 -100 -50 0 50 100 150 200

"PNG_st.dat"

t=0

t=1

t=2

t=3

t=4

xy

t

r

1

2

4

5

6

8

3

7

T

0

h = 3

1 2 3 4 5 6 7 8

6 4 7 1 3 8 2 5

1 2 53 7 846

1 3 62 5 847

P Q

λ1 = 3

sh = λ =

10

Zero temperature free fermion for PNG

Multi-layer PNG model

-100

-50

0

50

100

150

200

-200 -150 -100 -50 0 50 100 150 200

"mPNG"

We can see T = 0 free fermion before getting formulas.

Shadow line construction

P

Q

11

TASEP and Schur measure

TASEP = totally asymmetric simple exclusion process

· · · ⇒

1 − r

⇒

1 − r

· · ·

-3 -2 -1 0 1 2 3N(t): Integrated current at (0, 1) upto time t from step i.c.

-

6

(1, 1)

(N,N)

· · ·

...

i

j Waiting time of ith hop of jth particle

wij on (i, j): geometrically distributed

G(N,N) = maxup-right paths from(1,1)to(N,N)

( ∑(i,j)

on a path

wi,j

)Directed polymer at T = 0

Time at which N th particle arrives at the origin

12

Last passage percolation• RSK correspondence ⇒ Schur measure, DPP, Fredholm det

P[G(N,N) ≤ u] =1

Z

∑λ,λ1≤u

sλ(a)sλ(b) = det(1 − K)

Thanks to Jacobi-Trudi formula, Schur measure is a

determinantal point process(DPP), i.e., all correlation

functions are determinants with a common kernel K.• 2000 Johansson

limt→∞

P[N(t) − Jt

ct1/3≥ −s

]= F2(s) = det(1−K2)L2(s,∞)

where F2 is GUE Tracy-Widom distribution and kernel K2 is

K2(x, y) =

∫R+

Ai(x + λ)Ai(y + λ)dλ

13

2. KPZ equation and T > 0 polymer

• Height function h = h(x, t), x ∈ R, t ∈ R+,

∂

∂th =

1

2

∂2

∂x2h +

1

2

(∂h

∂x

)2

+ η

where η = η(x, t) is the space time white noise.

2λt/δx

h(x,t)

• Cole-Hopf transformation: Z = Z(x, t) = eh(x,t)

∂

∂tZ =

1

2

∂2

∂x2Z + ηZ

Directed polymer at T > 0

14

Fredholm determinant with T > 0 kernel

2010 TS-Spohn, Amir-Corwin-Quastel

Laplace transform of Z with i.c. Z(x, 0) = δ(x)

E[exp(−Zet24

−(t/2)1/3s)] = det(1 − Kt)L2(R+)

where the kernel has the Fermi-Dirac factor (”T > 0 kernel”),

Kt(x, y) =

∫R

Ai(x + λ)Ai(y + λ)

1 + e(t/2)1/3(s−λ)

dλ

Easy to take t → ∞ limit to get F2.

Obtaining the Fredholm determinant formula is nontrivial.

Can we see T > 0 free fermion before getting formulas?

15

3. Discrete KPZ models and q-Whittaker measure

• Discrete KPZ models: ASEP, q-TASEP, sHS6VM, etc.

2011 Borodin, Corwin

By the branching rule of q-Whittaker function, these models

are related to the q-Whittaker measure.

• q-PushTASEP(2015 Matveev-Petrov): kth particle position

xk(t + 1) = xk(t) + Vk,t + Wk,t, for k = 1, . . . , N,

where Vk,t ∼ qPo(akbt+1) and Wk,t whose distribution

depends on xk−1(t + 1) − xk−1(t), xk−1(t) − xk(t) − 1.

The N th particle position at time M is related to µ1 as

XN(M)d= µ1 + N .

16

Fredholm determinants for q-Whittaker

• Standard approach (2011- Borowin, Corwin, TS, Petrov, ...)

Markov duality + Bethe ansatz or by Macdonald operators.

E[

1

(ζq−µ1; q)∞

]= det(1 − K)

NOT T > 0 kernel. Asymptotics is rather involved.

Generalization to half-space case is difficult.

• Frobenius determinant approach (2019 Imamura, TS)

E[

1

(ζq−µ1; q)∞

]= det (1 − fζK)ℓ2(Z)

where fζ(m) = −ζqm

1−ζqm (T > 0 kernel!)

17

and

K(m1,m2) =

∫C

dz

z

∫D

dw

wga,b(z, w;m1,m2)

ga,b(z, w;m1,m2) =wm2

zm1

w

z − w

N∏i=1

(aiz; q)∞

(aiw; q)∞·

N∏j=1

(bj/w; q)∞

(bj/z; q)∞,

<z

=z

C

D

D

0 bmax...bmin

bmaxq...

1amax

1amin

· · · 1amaxq

...

18

Periodic Schur measure

• Periodic Schur measure (2007 Borodin, 2018 Betea-Bouttier)

1

Z

∑ρ∈P,ρ(⊂λ)

q|ρ|sλ/ρ(a)sλ/ρ(b)

• Its shift mix version is a DPP and hence

P (λ1 + S ≤ k) = det (1 + fL)ℓ2(Z)

where

f(m) = fζ(m)|ζ=−tq1/2+k =tq1/2+k+m

1 + tq1/2+k+m

L(m1,m2) =

∫C

dz

z

∫D

dw

wga,b(z, w,m1,m2).

19

Equivalence of the two Fredholm determinants

Combining all the above arguments, our identity can be written as

an identity between two Fredholm determinants.

Theorem

det (1 − fK)ℓ2(Z) = det (1 + fL)ℓ2(Z)

The only difference between the kernels K,L is the contours. The

equivalence can be proved by shift of contours.

This completes the first proof of our identity.

20

4. Bijective proof of the identity

Using Qµ = bµPµ with

bµ(q) =∏i≥1

1

(q; q)µi−µi+1

the identity can be written as

n∑ℓ=0

qℓ

(q; q)ℓ

∑µ:µ1=n−ℓ

bµ(q)Pµ(a)Pµ(b) =∑

λ,ρ:λ1=n

q|ρ|sλ/ρ(a)sλ/ρ(b)

21

Combinatorial formulas

Skew Schur function

sλ/ρ(x) =∑

T∈SST(λ/ρ)

xT

where SST is the set of skew semistandard tableaux.

q-Whittaker function (2000 Sanderson, 2012 Schling-Tingley)

Pµ(x) =∑

V ∈VST(µ)

qH(V )xV

where VST is the set of ”vertically strict tableaux” with increasing

elements in each column and no condition among columns and H

is the energy function.

22

Bijection Υ : (P,Q) ↔ (V,W, κ, ν)

(P,Q): A pair of skew SSTs with same shape λ/ρ

(V,W ): A pair of VSTs with same shape µ

κ ∈ K(µ) = {κ = (κ1, . . . , κµ1) ∈ Nµ10 : κi ≥ κi+1 if µ′

i = µ′i+1}

ν: partition

Weight preserving property

|ρ| = H(V ) + H(W ) + |κ| + |ν|

Note∑

κ∈K(µ) q|κ| = bµ(q) and P[ν1 = ℓ] = qℓ

(q;q)ℓ(q; q)∞.

23

Skew RSK dynamics

Iterated skew RSK maps by Sagan-Stanley on a periodic cylinder

(P,Q) =

(P2, Q2) =

(P10, Q10) =

Similar to Box-Ball systems! Crystals

24

5. New approach to KPZ models

• Standard approach to KPZ models with q has been to apply

Markov duality & Bethe ansatz or Macdonald operator

• With our bijection, one does not need to use such methods

any more. Once the mapping to (periodic) Schur measure is

established, then one can simply apply the methods of DPP

to get Fredholm determinants, for which asymptotic analysis

by now can be done in a standard way.

• Our approach also works also for half-space models.

25

Log Gamma polymer

Finite temperature directed polymer model (2009 Seppalainen)

-

6

(1, 1)

(N,N)

· · ·

...

i

jwij on (i, j): Gamma distributed

with parameter αi − αj

Prob for a path π from (1, 1) to (N,N)

1ZN

∏(i,j)∈π wi,j

• Can be studied by taking q → 1 limit of q-PushTASEP.

• Continuous limit ⇒ KPZ equation

26

Half space case

Half space q-Whittaker measure (2021 Imamura-Mucciconi-TS)

1

Φ(a, z; q)bµ(q; z)Pµ(a, q

2)

with Φ(a, z; q) =n∏

i=1

1

(aiz; q)∞

∏1≤i<j≤n

1

(aiaj; q2)∞

Here

bµ(q; z) =∏

i=2,4,6...

[qz2 + 1]µi−µi+1

q2

(q2; q2)µi−µi+1

∏i=1,3,5,...

z1µi>µi+1

(q; q)µi−µi+1

,

[A + B]kp =k∑

j=0

AjBk−j

(k

j

)p

,

(k

j

)p

=(p; p)k

(p; p)j(p; p)k−j.

27

Free boundary Schur measure• Free boundary Schur measure (2017 B-B-Nejjar-Vuletic )

1

Zzoddµ1 zoddλ

2 q|µ|sλ/µ(a) (Pfaffian point process)

• Thm:Identity relating HS q-Whittaker & FB Schur measuresk∑

ℓ=0

gℓ(γ, q)∑

µ:µ1=k−ℓ

bµ(q; γ)Pµ(a; q2) =

∑λ,ρ:λ1=k

γodd(λ′)+odd(ρ′)q|ρ|sλ/ρ(a)

with gk(γ, q) =[qγ2 + q2]kq2

(q2; q2)k.

• Fredholm Pfaffian for half-space q-Whittaker measure

E[1/(−ζq−µ1−χ; q)∞] = Pf(J − K)

where χ is a certain indep. r.v. and J is an anti-sym. kernel.

⇒ Log-Gamma polymer and KPZ equation in half-space

28

Summary

• We presented a new approach to study KPZ models by

mapping them to free fermions at finite temperature.

• Most KPZ models are related to q-Whittaker measure. We

have found an identity which relates marginals of q-Whittaker

and periodic Schur measures. The latter is related to free

fermions at finite temperature and DPP. This allows us to

write a quantity of a KPZ model as Fredholm determinant

with T > 0 kernel, which admits straightforward asymptotics.

A bijective proof of the identity will be explained in detail by

Matteo Mucciconi next week.

• Our approach works also for half-space models with Pfaffian

structures.

29

Geometric q-PushTASEP

2015 Matveev-Petrov

xk(t): kth particle position at time t.

xk(t + 1) = xk(t) + Vk,t + Wk,t, for k = 1, . . . , N,

where Vk,t ∼ qPo(akbt+1) with

X ∼ qPo(α) ⇔ P[X = n] =αn

(q; q)k(α; q)∞

and Wk,t ∼ φq−1,qgapk(t)(• |xk−1(t + 1) − xk−1(t)) where

gapk(t) = xk−1(t) − xk(t) − 1

φq−1,qa(r|c) = qar(qa; q−1)c−r(q−1; q−1)c

(q−1; q−1)r(q−1; q−1)c−r.

30

Kernel for half-space model

K1,1(x, y) =

∮C

dz

zx+1

∮D

dw

wy+1F (z)F (w)κ1,1(z, w)

K1,2(x, y) = −K2,1(y, x) =

∮dz

zx+1

∮dw

w−y+1

F (z)

F (w)κ1,2(z, w)

K2,2(x, y) =

∮dz

z−x+1

∮dw

w−y+1

1

F (z)F (w)κ2,2(z, w)

where

F (z) =n∏

i=1

(ai/z; q2)∞

(aiz; q2)∞

31

κi,j

κ1,1 =1

ζz1/2w3/2

(q2; q2)2∞(qz, qw,−1

z,− 1

w; q)∞

θq2(w/z)

θq2(q2zw)

θ3(ζ2z2w2; q4)

θ3(ζ2, q4)g(z)g(w)

κ1,2 =w1/2

z1/2

(q2; q2)2∞(qz,−qw,−1/z, 1/w; q)∞

θq2(q2wz)

θq2(w/z)

θ3(ζ2z2/w2; q4)

θ3(ζ2, q4)

g(z)

g(w)

κ2,2 =ζ

z1/2w3/2

(q2; q2)2∞(−qz,−qw, 1

z, 1w; q)∞

θq2(w/z)

θq2(q2wz)

θ3(ζ2/z2w2; q4)

θ3(ζ2, q4)

1

g(z)g(w)

with

g(z) =(qz; q)∞

(1/z; q)∞

(γq/z, γ/z; q2)∞

(γqz, γq2z; q2)∞

1

(γ2q; q2)∞(−q; q)∞

32