connecting q-whittaker and periodic schur measures
TRANSCRIPT
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)∞
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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