new edge asymptotics of skew young diagrams via free...
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New edge asymptotics of skew Young diagrams via freeboundaries
Dan Betea
University of Bonn
joint work with J. Bouttier, P. Nejjar and M. Vuletic
FPSAC, Ljubljana, 2019
4.VI1.MM19
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Outline
This talk contains stuff on
I partitions and tableaux
I the Plancherel (mostly) and uniform measures on Young diagrams
I main results on skew Young diagrams
I the beyond
and a few surprises.
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Partitions
• • ◦ ◦ ◦ ◦ ◦•◦••◦••
Figure: Partition (Young diagram) λ = (2, 2, 2, 1, 1) (Frobenius coordinates (1, 0|4, 1)) in English, French and Russian notation, with
associated Maya diagram (particle-hole representation). Size |λ| = 8, length `(λ) = 5.
Figure: Skew partitions (Young diagrams) (4, 3, 2, 1)/(2, 1) (but also (5, 4, 3, 2, 1)/(5, 2, 1), . . . ) and (4, 4, 2, 1)/(2, 2) (but also
(6, 4, 4, 2, 1)/(6, 2, 2), . . . )
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Counting tableaux
A standard (semi-standard) Young tableau SYT (SSYT) is a filling of a (possibly skew)Young diagram with numbers 1, 2, . . . strictly increasing down columns and rows (rowsweakly increasing for semi-standard).
1 3 5 62 4 978
1 1 2 22 2 334
1 73 4
2 56
1 21 3
2 23
dimλ := number of SYTs of shape λ,
dimλ := number of SSYTs of shape λ with entries from 1 . . . n
and similarly for dimλ/µ, dimλ/µ.
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Two natural measures on partitions
I On partitions of n (|λ| :=∑λi = n): Plancherel vs. uniform
Prob(λ) =(dimλ)2
n!vs. Prob(λ) =
1
#{partitions of n}
I On all partitions: poissonized Plancherel vs. (grand canonical) uniform
Prob(λ) = e−ε2ε2|λ| (dimλ)2
(|λ|!)2vs. Prob(λ) = u|λ|
∏i≥1
(1− ui )
with ε > 0, 1 > u > 0 parameters.
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Ulam’s problem and Hammersley last passage percolation I
PPP(ε2) in the unit square.
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Ulam’s problem and Hammersley last passage percolation II
Quantity of interest: L = longest up-right path from (0, 0) to (1, 1) (= 4 here).
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Ulam’s problem and Hammersley last passage percolation III
1 2 3 4 5 6 7 8 109
1
2
3
8
9
10
4
5
6
7
9 4 7 2 5 8 6 1 310
L is the length (any) of the longest increasing subsequence in a random permutation ofSN with N ∼ Poisson(ε2).
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The poissonized Plancherel measure
By the Robinson–Schensted–Knuth correspondence and Schensted’s theorem, L = λ1 indistribution where λ has the poissonized Plancherel measure:
Prob(λ) = e−ε2ε2|λ| (dimλ)2
(|λ|!)2
= e−ε2sλ(plε)sλ(plε)
(s is a Schur function, plε the Plancherel specialization sending p1 → ε, pi → 0, i ≥ 2)
Interest: what happens to λ1 as ε→∞? (large PPP, large random permutation, ...)
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Limit shape
A Plancherel-random representation (partition!) of S2304 (Prob(λ) = (dimλ)2/n!,n = 2304), at IHP. The limit shape should be obvious (VerKer, LogShe 1977).
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Limit shapes: Plancherel vs uniform
Random Plancherel (left) and uniform (right) partitions of N = 10000. The scale is
different:√N for Plancherel,
√N log N for uniform.
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The Baik–Deift–Johansson theorem and Tracy–Widom
Theorem (BaiDeiJoh 1999)If λ is distributed as poissonized Plancherel, we have:
limε→∞
Prob
(λ1 − 2ε
ε1/3≤ s
)= FGUE(s) := det(1− Ai2)L2(s,∞)
with
Ai2(x , y) :=
∫ ∞0
Ai(x + s)Ai(y + s)ds
and Ai the Airy function (solution of y ′′ = xy decaying at ∞).
FGUE is the Tracy–Widom GUE distribution. It is by (original) construction the extremedistribution of the largest eigenvalue of a random hermitian matrix with iid standardGaussian entries as the size of the matrix goes to infinity.
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The Erdos–Lehner theorem and Gumbel
Theorem (ErdLeh 1941)For the uniform measure Prob(λ) ∝ u|λ| we have:
limu→1−
Prob
(λ1 < −
log(1− u)
log u+
ξ
| log u|
)= e−e−ξ .
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The finite temperature Plancherel measure
On pairs of partitions µ ⊂ λ ⊃ µ consider the measure (Bor 06)
Prob(µ, λ) ∝ u|µ| ·ε|λ|−|µ| dim2(λ/µ)
(|λ/µ|!)2
with u = e−β , β = inverse temperature.
I u = 0 yields the poissonized Plancherel measure
I ε = 0 yields the (grand canonical) uniform measure
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What is in a part?
PPP (ε2)
PPP (uε2)
PPP (u2ε2)
PPP (u3ε2)
PPP (u4ε2)
With L the longest up-right path in this cylindric geometry, in distribution, Schensted’stheorem states that
λ1 = L + κ1
where κ is a uniform partition Prob(κ) ∝ u|κ| independent of everything else.
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The finite temperature Plancherel measure II
Theorem (B/Bouttier 2019)Let M =
√ε
1−u→∞ and u = exp(−αM−1/3)→ 1. Then
limM→∞
Prob
(λ1 − 2M
M1/3≤ s
)= Fα(s) := det(1− Aiα)L2(s,∞)
with
Aiα(x , y) :=
∫ ∞−∞
eαs
1 + eαs· Ai(x + s)Ai(y + s)ds
the finite temperature Airy kernel.
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A word on the finite temperature Airy kernel
Aiα is Johansson’s (2007) Airy kernel in finite temperature (also appearing as the KPZcrossover kernel: SasSpo10 and AmiCorQua11, in random directed polymersBorCorFer11, cylindric OU processes LeDMajSch15):
Aiα(x , y) =
∫ ∞−∞
eαs
1 + eαsAi(x + s)Ai(y + s)ds
and interpolates between the Airy kernel and a diagonal exponential kernel:
limα→∞
Aiα(x , y) = Ai2(x , y),
limα→0+
1
αAiα
(x
α−
1
2αlog(4πα3),
y
α−
1
2αlog(4πα3)
)= e−xδx,y .
If Fα(s),FGUE(s), and G(s) are the Fredholm determinants on (s,∞) of Aiα,Ai2 ande−xδx,y , then (Joh 2007)
limα→∞
Fα(s) = FGUE(s), limα→0+
Fα(
s
α−
1
2αlog(4πα3)
)= G(s) = e−e−s
.
It appeared in seemingly two different situations:
I random matrix models on the cylinder/in finite temperature (Joh, LeDMajSch, ...)
I the KPZ equation with wedge I.C. at finite time (SasSpo, AmiCorQua, ...)
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Three limiting regimes for edge fluctuations
Theorem (B/Bouttier 2019)With u = e−r → 1 as r → 0+ and ε→∞ (or finite) we have:
I εr2 → 0+ leads to Gumbel behavior; thermal fluctuations win
I εr2 →∞ leads to Tracy–Widom; quantum fluctuations win
I εr2 → α ∈ (0,∞) leads to finite temperature Tracy–Widom Fα; equilibriumbetween thermal and quantum
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The stuff that’s in the FPSAC abstract
Consider the following measures (oc = number of odd columns, n letters for dim):
M↗(µ, λ) ∝ aoc(µ)1 a
oc(λ)2 · u|µ| ·
ε|λ/µ| dim(λ/µ)
|λ/µ|!,
M↗↘(µ, λ, ν) ∝ aoc(µ)1 a
oc(λ)2 · u|µ|v |ν| ·
ε|λ/µ|+|λ/ν| dim(λ/µ) dim(λ/ν)
|λ/µ|! · |λ/ν|!,
M↗(µ, λ) ∝ aoc(µ)1 a
oc(λ)2 · u|µ| · q|λ/µ| · dim(λ/µ),
M↗↘(µ, λ, ν) ∝ aoc(µ)1 a
oc(λ)2 · u|µ|v |ν| · q|λ/µ|+|λ/ν| · dim(λ/µ)dim(λ/ν).
They all interpolate between Plancherel-type (u = 0) and uniform (ε, q = 0) measures.
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What is in a part? (λ1 = L + κ1 via RSK)
0
0
0
0
0 0
0
0
0
0
00000
0
00
0
0
0
0
0
0
0
0
7
15
1310
8
711
98
10
8 0
0
00
0
0
00
0
1
1
1
1
25
5
5
5
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
1
1
1
1
1
1
1
1
6
6
6
3
3
3
3
14
17
3
3
3
4
4
4
4
4
2
2
2
23
01
1
1
1
0
0
0
0
0
2
2
2
2
3
y4y3y2y1
x4
x3
x2
x1
Geom(x3y2)
Geom((uv)2x2y3)
Geom((uv)4x3y2)
Geom(v2y1y2)
Geom(u2(uv)2x1x2)
Geom(vy3)
Geom(u(uv)x3)
Geom(x2y4)
Geom((uv)2x4y2)
Geom((uv)4x2y4)
Geom(u2x2x4)
Geom(v2(uv)2y2y4)
Geom(ux2)
Geom(v(uv)y2)
κκκκκ
κ
κ
κ
κ
µλ
ν
0
0
0
0
0 0
000
0
0
0
0
0
7
1
8
2
9
2
10
0
0
9
0
00
11
1
11
1
25
5
2
2
2
12
1
1
1
18
0
3
3
33
4
4
4
3
2
2
2
0
1
1
0
7
10
3
2
2
3
y4y3y2y1
Geom(y2y4)
Geom(u4y2y4)
Geom(u8y2y4)
Geom(u2y2y4)
Geom(u6y2y4)
Geom(uy2)
Geom(u2y2)
κ
κ
κ
κ
κ
µλ
Geom(y3)
Geom(uy3)
Geom(u2y3)
Figure: Longest up-right path in orange of length L = 199 (left) and L = 130 (right). M↗↘(µ, λ, ν) (left) and M↗(µ, λ) (right);xi = yi = q; case a1 = a2 = 0 (for generic, multiply the parameters in the boundary triangles by a1 and a2 for the two different
boundaries; κ is uniform with prob.∝ (uv)|κ| (left) and ∝ u|κ| (right).
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Main theorem: edge limits (SYT case)
Theorem (B/Bouttier/Nejjar/Vuletic FPSAC 2019)Fix η, αi , i = 1, 2 positive reals. Let M := ε
1−u2 →∞ and set
u = v = exp(−ηM−1/3), ai = uαi/η , i = 1, 2
all going to 1 as M →∞. (In particular, ε ∼ M2/3 →∞.) We have:
limM→∞
M↗(λ1 − 2M
M1/3≤ s +
1
ηlog
M1/3
η
)= F 1;α1,α2;η(s),
limM→∞
M↗↘(λ1 − 2M
M1/3≤ s +
1
2ηlog
M1/3
2η
)= F 2;α1,α2;η(s)
with the distributions F ··· explicit Fredholm pfaffians.
Remark: This theorem generalizes celebrated results of Baik–Rains (2000) on longestincreasing subsequences in symmetrized permutations, as well as the classicalBaik–Deift–Johansson theorem.
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Main theorem: edge limits (SSYT case on n letters)
Theorem (B/Bouttier/Nejjar/Vuletic FPSAC 2019)Fix η, αi , i = 1, 2 positive reals. As n→∞ (n a positive integer), let
u = v = exp(−ηn−1/3), ai = uαi/η , i = 1, 2
all going to 1 and set q = 1− u2 → 0. We have:
limn→∞
M↗(λ1 − χnn1/3
≤ s +1
ηlog
n1/3
η
)= F 1;α1,α2;η(s),
limn→∞
M↗↘(λ1 − χnn1/3
≤ s +1
2ηlog
n1/3
2η
)= F 2;α1,α2;η(s)
where χ = 2q∑`≥0
u2`
1−u2`q.
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Limits to Tracy–Widom
Theorem (B/Bouttier/Nejjar/Vuletic FPSAC 2019)We have:
limη→∞
F 1;α1,α2;η(s) = F�(s;α2), limη→∞
F 2;α1,α2;η(s) = FGUE(s)
where FGUE is the Tracy–Widom GUE distribution and F�(s;α2) is the Baik–RainsTracy–Widom GOE/GSE crossover
F�(s; 0) = FGOE(s), F�(s;∞) = FGSE(s).
Remark: as η → 0, the distributions should converge to Gumbel in the appropriate (sofar unknown) scaling.
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Defition of distribution functions
The distributions are Fredholm pfaffians F k;α1,α2;η(s) = pf(J − Ak;α1,α2;η
)L2(s+ log 2
k·η ,∞)
for specific 2× 2 matrix kernels A. For example:
A1;α1,α2;η1,1
(x, y) =
∫ ∫Γ
(ζ
η,ω
η
)γ
(1)(ζ)γ(1)(ω)sinπ(ζ−ω)
2η
sinπ(ζ+ω)
2η
eζ3
3−xζ+ω
3
3−yω
dζω,
A1;α1,α2;η1,2
(x, y) =
∫ ∫Γ
(ζ
η, 1 −
ω
η
)γ(1)(ζ)
γ(1)(ω)
sinπ(ζ+ω)
2η
sinπ(ζ−ω)
2η
eζ3
3−xζ−ω
3
3+yω dζω
2η
= − A1;α1,α2;η2,1
(y, x),
A1;α1,α2;η2,2
(x, y) =
∫ ∫Γ
(1 −
ζ
η, 1 −
ω
η
)1
γ(1)(ζ)γ(1)(ω)
sinπ(ζ−ω)
2η
sinπ(ζ+ω)
2η
e− ζ
3
3+xζ−ω
3
3+yω dζω
4η2
− sgn(x − y)
where dζω =dζdω
(2πi)2 , γ(1)(ζ) :=
Γ
(12
+α1−ζ
2η,1+
α2−ζ2η
)Γ
(12
+α1+ζ
2η,α2+ζ
2η
) , Γ(a, b, c, . . . ) = Γ(a)Γ(b)Γ(c) · · · and where the
contours are certain top-to-bottom vertical lines close enough to 0.
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When α1 = α2 = 0 (no boundary parameters) things simplify
A1;η1,1
(x, y) =
∫ ∫Γ
(1 −
ζ
η, 1 −
ω
η
)sinπ(ζ−ω)
2η
sinπ(ζ+ω)
2η
eζ3
3−xζ+ω
3
3−yω dζω
4,
A1;η1,2
(x, y) =
∫ ∫Γ
(1 −
ζ
η,ω
η
)sinπ(ζ+ω)
2η
sinπ(ζ−ω)
2η
eζ3
3−xζ−ω
3
3+yω dζω
2η= −A
1;η2,1
(y, x),
A1;η2,2
(x, y) =
∫ ∫Γ
(ζ
η,ω
η
)sinπ(ζ−ω)
2η
sinπ(ζ+ω)
2η
e− ζ
3
3+xζ−ω
3
3+yω dζω
η2
+
∫Γ
(ζ
η
)e− ζ
3
3+xζ dζ
η−∫
Γ
(ω
η
)e−ω
3
3+yω dω
η− sgn(x − y);
A2;η1,1
(x, y) =
∫ ∫Γ
(1
2−
ζ
2η,
1
2−ω
2η
)sinπ(ζ−ω)
4η
cosπ(ζ+ω)
4η
eζ3
3−xζ+ω
3
3−yω dζω
4η,
A2;η1,2
(x, y) =
∫ ∫Γ
(1
2−
ζ
2η,
1
2+ω
2η
)cos
π(ζ+ω)4η
sinπ(ζ−ω)
4η
eζ3
3−xζ−ω
3
3+yω dζω
4η= −A
2;η2,1
(y, x),
A2;η2,2
(x, y) =
∫ ∫Γ
(1
2+ζ
2η,
1
2+ω
2η
)sinπ(ζ−ω)
4η
cosπ(ζ+ω)
4η
e− ζ
3
3+xζ−ω
3
3+yω dζω
4η.
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Proof
I pass to the grand canonical ensemble by introducing an independent (even) charge
2d from Prob(d) ∝ t2d (uv)2d2shifting every part in every partition
I rewrite measures in terms of skew Schur functions, for example
M↗↘ext (µ, λ, ν, d) ∝ t2d (uv)2d2· aoc(µ)
1 aoc(λ)2 · u|µ|v |ν| · sλ/µ(q, . . . , q)sλ/ν(q, . . . , q)
I rewrite in terms of lattice (g`∞ free) fermions and use new Wick lemma to obtainpfaffian correlations for the point process
I steepest descent analysis of correlation kernel
I remove charge at the end
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Conclusion
Moral of the story: natural combinatorial measures on integer partitions lead tointeresting asymptotic probabilistic behavior.
Future directions:
I Universality of the limiting distributions
I Connections to integrable hierarchies (i.e. the universal character hierarchy)
I Relation to (recent) work on asymptotics of dimλ/µ
I Connections to (asymptotic) representation theory (the Okounkov–Olshanskiformula for dimλ/µ)
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Thank you!