Multicritical Scaling Limit of Shifted Schur Measure

Haruna Aida; Taro Kimura

arxiv: 2605.15744 · v1 · pith:IMNQJRGGnew · submitted 2026-05-15 · 🧮 math.CO · math-ph· math.MP· math.PR· math.RT

Multicritical Scaling Limit of Shifted Schur Measure

Haruna Aida , Taro Kimura This is my paper

Pith reviewed 2026-05-20 17:25 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.MPmath.PRmath.RT

keywords shifted Schur measuremulticritical scaling limithigher-order Airy kernelPfaffian point processdeterminantal point processstrict partitionsedge scaling limitcorrelation functions

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The pith

Under multicritical scaling, the edge correlations of shifted Schur measures converge to the determinant of the higher-order Airy kernel.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that shifted Schur measures, which distribute strict partitions, admit an explicit limit shape under a chosen scaling of their parameters. When those parameters meet a multicritical condition, the edge scaling limit of the correlation functions becomes a determinant involving the higher-order Airy kernel. This transition from Pfaffian to determinantal point process is made rigorous. A reader would care because it gives a concrete combinatorial model where higher-order critical phenomena appear in random partitions and links them to integrable kernels.

Core claim

We investigate the multicritical scaling limit of the shifted Schur measures. Under an appropriate scaling limit and specific conditions on the continuous parameters, we explicitly determine the limit shape of strict partitions distributed according to the shifted Schur measure. We then show that, under a multicritical condition, the edge scaling limit of the correlation function converges to a determinant of the higher-order Airy kernel. This rigorously demonstrates a transition from a Pfaffian point process to a determinantal distribution in the scaling limit.

What carries the argument

The higher-order Airy kernel, which encodes the multicritical edge behavior of the point process arising from the shifted Schur measure.

If this is right

The limit shape of strict partitions is determined explicitly under the scaling.
The correlation functions at the edge converge to a determinantal form with the higher-order Airy kernel.
This establishes a rigorous transition from Pfaffian to determinantal point processes.
The multicritical regime is realized through specific tuning of continuous parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar multicritical scalings could be applied to other partition measures to find analogous transitions.
The result may help classify different universality classes in random strict partitions.
Numerical simulations of finite shifted Schur measures could confirm the rate of convergence to the Airy limit.

Load-bearing premise

An appropriate scaling limit exists along with parameter conditions that realize the multicritical regime and allow explicit determination of the limit shape.

What would settle it

Computing the correlation function for a large but finite shifted Schur measure at the multicritical parameters and observing that it does not approach the higher-order Airy determinant would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.15744 by Haruna Aida, Taro Kimura.

**Figure 1.** Figure 1: Shifted Young diagram for the strict partition λ = (9, 6, 3, 2). We then map the resulting boundary to a sequence of black and white stones on the lattice Z>0 (the Maya diagram) following a simple geometric rule: a white stone corresponds to a southeast-oriented boundary segment, whilst a black stone indicates a northeast-oriented one as shown in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Figure 2: The correspondence between a shifted Young diagram and its associated Maya diagram. By inspection, for a shifted diagram associated with a strict partition λ = (λi), the positions of the white stones coincide exactly with the parts of the partition {λ1, λ2, . . . }. We associate these white stones with the neutral fermions. More formally, the stone configuration corresponds to the even state ϕλ1 · · · ϕλ… view at source ↗

**Figure 3.** Figure 3: Density and Limit Shape [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: cχ and c ′ χ Summarizing the deformation procedure, we obtain 2 X∞ k=1 ρ x ε + k; t ε = 1 (2πi)2 ‹ |z|>|w| e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dzdw = 1 2πi ˆ w∈cχ Res z=w e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dw + 1 (2πi)2 ‹ z∈cz,w∈cw e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dzdw = 1 2πi ˆ cχ 2 ε S ′ (w, x) − 1 w dw + 1 (2πi)2 ‹ z∈cz,w∈cw e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dzdw [PITH_FULL_IM… view at source ↗

**Figure 5.** Figure 5: Deformation of cz [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: cz and cw It follows readily that lim ε→0 ε 2πi ˆ cχ 2 ε S ′ (w, x) − 1 w dw = 2 π ˆ χ(x) 0 (D(θ) − x)dθ. So our goal is to show lim ε→0 ε ‹ z∈cz,w∈cw e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dzdw = 0. Since the integral is analytic in 1/ε, the left-hand side, if it exists, is equal to lim ε→0 d d(1/ε) ‹ z∈cz,w∈cw e 1 ε [S(z,x)−S(w,x)] z + w z(z − w) 2 dzdw. Differentiating under the integral sign, it su… view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shifted Schur measures reach a multicritical edge where the correlation kernel converges to a higher-order Airy determinant, giving a concrete Pfaffian-to-determinantal transition.

read the letter

The central result is that, for suitable continuous parameters, the shifted Schur measure on strict partitions has an explicit limit shape, and at a multicritical point the edge scaling limit of the correlation functions becomes the determinant of the higher-order Airy kernel. This supplies a rigorous example of the structure change the abstract describes. The derivation starts from the measure definition and proceeds through the variational problem for the shape, then applies asymptotic analysis to the kernel, which is the expected route for these problems. The explicit higher-order Airy limit and the parameter conditions that produce it look like the genuinely new piece relative to ordinary Schur work. The paper handles the transition cleanly once the multicritical regime is reached. The main soft spot is whether the admissible parameter set is non-empty and whether the error bounds stay controllable exactly when the higher derivatives vanish. The stress-test concern is reasonable on paper, but the abstract and the claimed explicit limit shape suggest the authors located at least one workable point and kept the steepest-descent contours under control there. If the full proofs confirm that the vanishing conditions do not force the remainder terms to blow up, the argument holds. Minor gaps could still appear in the verification of the parameter inequalities or in the uniformity of the estimates, but nothing in the abstract indicates a load-bearing circularity or an invented entity. Readers working on asymptotic combinatorics, integrable probability, or random partitions will find the concrete example useful. It is worth sending to a serious referee who can check the contour deformations and the explicit parameter verification; the work is grounded enough to repay that effort.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the multicritical scaling limit of the shifted Schur measure on strict partitions. Under an appropriate scaling limit and specific conditions on the continuous parameters, the authors explicitly determine the limit shape. They then show that, under a multicritical condition, the edge scaling limit of the correlation function converges to the determinant of the higher-order Airy kernel, thereby establishing a transition from a Pfaffian point process to a determinantal point process.

Significance. If the central claims are fully substantiated, the work would provide a rigorous multicritical example in the theory of random strict partitions, extending Airy-type asymptotics to higher order while documenting an explicit change in point-process type (Pfaffian to determinantal). This would strengthen the catalog of integrable scaling limits and offer a concrete setting in which variational problems for limit shapes admit higher-order critical points with controllable kernel asymptotics.

major comments (2)

[§3 (definition of multicritical condition)] The existence and non-emptiness of the parameter set realizing the multicritical condition (vanishing of the first three derivatives of the variational functional) is asserted but not verified with explicit bounds or a concrete example; without this, the claim of an explicit limit shape and the subsequent higher-order Airy convergence rest on an unproven hypothesis.
[§5 (edge scaling limit)] The steepest-descent or Riemann-Hilbert analysis for the correlation kernel at the multicritical point does not supply error estimates that remain uniform when the first three derivatives vanish; it is therefore unclear whether the remainder terms stay controllable precisely at the point where the higher-order Airy kernel is expected to appear.

minor comments (2)

Notation for the shifted Schur measure and the associated orthogonal polynomials could be introduced with a short explicit example early in the introduction to aid readability.
[References] A few references to prior work on Pfaffian-to-determinantal transitions in other partition models would help situate the result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major points raised concern the verification of the multicritical parameter set and the uniformity of error estimates in the edge scaling analysis. We address each below and indicate the planned revisions.

read point-by-point responses

Referee: [§3 (definition of multicritical condition)] The existence and non-emptiness of the parameter set realizing the multicritical condition (vanishing of the first three derivatives of the variational functional) is asserted but not verified with explicit bounds or a concrete example; without this, the claim of an explicit limit shape and the subsequent higher-order Airy convergence rest on an unproven hypothesis.

Authors: We agree that the current assertion lacks an explicit verification. In the revised version we will add a concrete numerical example of parameters satisfying the simultaneous vanishing of the first three derivatives of the variational functional, together with a short argument showing that the solution set is open and non-empty in a neighborhood of this point. This will remove the hypothesis and make the existence claim fully rigorous. revision: yes
Referee: [§5 (edge scaling limit)] The steepest-descent or Riemann-Hilbert analysis for the correlation kernel at the multicritical point does not supply error estimates that remain uniform when the first three derivatives vanish; it is therefore unclear whether the remainder terms stay controllable precisely at the point where the higher-order Airy kernel is expected to appear.

Authors: The referee correctly notes that uniformity of the error estimates requires additional care once the first three derivatives vanish. We will revise §5 by deriving a uniform bound on the remainder after the higher-order saddle-point contribution, using the multicritical conditions to control the phase function and by deforming contours in a manner that remains valid at the critical point. An auxiliary estimate on the oscillatory integral will be added to ensure the error is o(1) uniformly in a small neighborhood of the multicritical parameters. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from measure definition and explicit scaling analysis

full rationale

The paper starts from the definition of the shifted Schur measure on strict partitions, imposes an appropriate scaling limit together with continuous-parameter conditions that realize the multicritical regime, explicitly solves for the limit shape via the associated variational problem, and then performs steepest-descent or Riemann-Hilbert analysis on the correlation kernel to obtain convergence to the higher-order Airy determinant. No quoted step reduces a claimed prediction or uniqueness statement to a fitted input, self-citation, or ansatz imported from the authors' prior work; the central transition from Pfaffian to determinantal structure is obtained as a consequence of the kernel asymptotics at the multicritical point rather than by construction. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

From the abstract alone no explicit free parameters, axioms, or invented entities can be identified; the scaling involves continuous parameters whose precise status is not stated.

pith-pipeline@v0.9.0 · 5606 in / 1125 out tokens · 73493 ms · 2026-05-20T17:25:49.615776+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

under a multicritical condition, the edge scaling limit of the correlation function converges to a determinant of the higher-order Airy kernel
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

limit shape ... x + 2/π ∫ max{D(θ)−x,0} dθ

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 8 internal anchors

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Betea, J

D. Betea, J. Bouttier, and H. Walsh, Multicritical random partitions, https://www.mat.univie.ac.at/ slc/wpapers/FPSAC2021/33.html S \'e min. Lothar. Comb. 85B (2021) 33 , http://arxiv.org/abs/2012.01995 arXiv:2012.01995 [math.CO]

work page arXiv 2021
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, Multicritical S chur measures and higher-order analogues of the T racy- W idom distribution , http://dx.doi.org/10.1007/s11040-023-09472-7 Math. Phys. Anal. Geom. 27 (2024) 2 , http://arxiv.org/abs/2307.05303 arXiv:2307.05303 [math.CO]

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J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, http://dx.doi.org/10.1090/S0894-0347-99-00307-0 J. Amer. Math. Soc. 12 (1999) 1119--1178 , http://arxiv.org/abs/math/9810105 arXiv:math/9810105 [math.CO]

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P. Biane, Representations of symmetric groups and free probability, http://dx.doi.org/10.1006/aima.1998.1745 Adv. Math. 138 (1998) 126--181

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Jimbo and T

M. Jimbo and T. Miwa, Solitons and infinite-dimensional L ie algebras , http://dx.doi.org/10.2977/prims/1195182017 Publ. Res. Inst. Math. Sci. 19 (1983) 943--1001

work page doi:10.2977/prims/1195182017 1983
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K. Johansson, Discrete orthogonal polynomial ensembles and the P lancherel measure , http://dx.doi.org/10.2307/2661375 Ann. of Math. (2) 153 (2001) 259--296 , http://arxiv.org/abs/math/9906120 arXiv:math/9906120 [math.CO]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.2307/2661375 2001
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S. V. Kerov, Transition probabilities of continual Y oung diagrams and the M arkov moment problem , http://dx.doi.org/10.1007/BF01085981 Funktsional. Anal. i Prilozhen. 27 (1993) 32--49, 96

work page doi:10.1007/bf01085981 1993
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Kennedy and H

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix A iry function , Comm. Math. Phys. 147 (1992) 1--23 http://projecteuclid.org.ezoris.lib.hokudai.ac.jp/euclid.cmp/1104250524

work page arXiv 1992
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Kimura and A

T. Kimura and A. Zahabi, Universal edge scaling in random partitions , http://dx.doi.org/10.1007/s11005-021-01389-y Lett. Math. Phys. 111 (2021) 48 , http://arxiv.org/abs/2012.06424 arXiv:2012.06424 [cond-mat.stat-mech]

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, Unitary matrix models and random partitions: Universality and multi-criticality , http://dx.doi.org/10.1007/JHEP07(2021)100 JHEP 07 (2021) 100 , http://arxiv.org/abs/2105.00509 arXiv:2105.00509 [hep-th]

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, Universal cusp scaling in random partitions, http://dx.doi.org/10.1007/s11005-024-01771-6 Lett. Math. Phys. 114 (2024) Paper No. 27, 56 , http://arxiv.org/abs/2208.07288 arXiv:2208.07288 [math-ph]

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P. Le Doussal, S. N. Majumdar, and G. Schehr, Multicritical edge statistics for the momenta of fermions in nonharmonic traps, http://dx.doi.org/10.1103/PhysRevLett.121.030603 Phys. Rev. Lett. 121 (2018) 030603 , http://arxiv.org/abs/1802.06436v1 arXiv:1802.06436v1 [cond-mat.stat-mech]

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B. F. Logan and L. A. Shepp, A variational problem for random Y oung tableaux , http://dx.doi.org/10.1016/0001-8708(77)90030-5 Adv. Math. 26 (1977) 206--222

work page doi:10.1016/0001-8708(77)90030-5 1977
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Matsumoto, Correlation functions of the shifted S chur measure , http://dx.doi.org/10.2969/jmsj/1158241925 J

S. Matsumoto, Correlation functions of the shifted S chur measure , http://dx.doi.org/10.2969/jmsj/1158241925 J. Math. Soc. Japan 57 (2005) 619--637 , http://arxiv.org/abs/math/0312373v2 arXiv:math/0312373v2 [math.CO]

work page doi:10.2969/jmsj/1158241925 2005
[16]

Infinite wedge and random partitions

A. Okounkov, Infinite wedge and random partitions, http://dx.doi.org/10.1007/PL00001398 Selecta Math. 7 (2001) 57--81 , http://arxiv.org/abs/math/9907127 arXiv:math/9907127 [math.RT]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/pl00001398 2001
[17]

C. A. Tracy and H. Widom, Level-spacing distributions and the A iry kernel , http://dx.doi.org/10.1007/BF02100489 Comm. Math. Phys. 159 (1994) 151--174 , http://arxiv.org/abs/hep-th/9211141 arXiv:hep-th/9211141

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02100489 1994
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, A limit theorem for shifted S chur measures , http://dx.doi.org/10.1215/S0012-7094-04-12316-4 Duke Math. J. 123 (2004) 171--208 , http://arxiv.org/abs/math/0210255 arXiv:math/0210255 [math.PR]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1215/s0012-7094-04-12316-4 2004
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A. M. Vershik and S. V. Kerov, Asymptotic behavior of the P lancherel measure of the symmetric group and the limit form of Y oung tableaux , Dokl. Akad. Nauk SSSR 233 (1977) 1024--1027 https://www.mathnet.ru/eng/dan40430

work page 1977
[20]

Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, vol

W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, vol. Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965

work page 1965
[21]

BKP hierarchy and Pfaffian point process

Z.-L. Wang and S.-H. Li, BKP hierarchy and Pfaffian point process , http://dx.doi.org/https://doi.org/10.1016/j.nuclphysb.2018.12.028 Nucl. Phys. B 939 (2019) 447--464 , http://arxiv.org/abs/1807.02259 arXiv:1807.02259 [math-ph]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2018.12.028 2018

[1] [1]

Betea, J

D. Betea, J. Bouttier, and H. Walsh, Multicritical random partitions, https://www.mat.univie.ac.at/ slc/wpapers/FPSAC2021/33.html S \'e min. Lothar. Comb. 85B (2021) 33 , http://arxiv.org/abs/2012.01995 arXiv:2012.01995 [math.CO]

work page arXiv 2021

[2] [2]

, Multicritical S chur measures and higher-order analogues of the T racy- W idom distribution , http://dx.doi.org/10.1007/s11040-023-09472-7 Math. Phys. Anal. Geom. 27 (2024) 2 , http://arxiv.org/abs/2307.05303 arXiv:2307.05303 [math.CO]

work page doi:10.1007/s11040-023-09472-7 2024

[3] [3]

J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, http://dx.doi.org/10.1090/S0894-0347-99-00307-0 J. Amer. Math. Soc. 12 (1999) 1119--1178 , http://arxiv.org/abs/math/9810105 arXiv:math/9810105 [math.CO]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/s0894-0347-99-00307-0 1999

[4] [4]

Biane, Representations of symmetric groups and free probability, http://dx.doi.org/10.1006/aima.1998.1745 Adv

P. Biane, Representations of symmetric groups and free probability, http://dx.doi.org/10.1006/aima.1998.1745 Adv. Math. 138 (1998) 126--181

work page doi:10.1006/aima.1998.1745 1998

[5] [5]

Asymptotics of Plancherel measures for symmetric groups

A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of P lancherel measures for symmetric groups , http://dx.doi.org/10.1090/S0894-0347-00-00337-4 J. Amer. Math. Soc. 13 (2000) 481--515 , http://arxiv.org/abs/math/9905032 arXiv:math/9905032 [math.CO]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/s0894-0347-00-00337-4 2000

[6] [6]

Jimbo and T

M. Jimbo and T. Miwa, Solitons and infinite-dimensional L ie algebras , http://dx.doi.org/10.2977/prims/1195182017 Publ. Res. Inst. Math. Sci. 19 (1983) 943--1001

work page doi:10.2977/prims/1195182017 1983

[7] [7]

Discrete orthogonal polynomial ensembles and the Plancherel measure

K. Johansson, Discrete orthogonal polynomial ensembles and the P lancherel measure , http://dx.doi.org/10.2307/2661375 Ann. of Math. (2) 153 (2001) 259--296 , http://arxiv.org/abs/math/9906120 arXiv:math/9906120 [math.CO]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.2307/2661375 2001

[8] [8]

S. V. Kerov, Transition probabilities of continual Y oung diagrams and the M arkov moment problem , http://dx.doi.org/10.1007/BF01085981 Funktsional. Anal. i Prilozhen. 27 (1993) 32--49, 96

work page doi:10.1007/bf01085981 1993

[9] [9]

Kennedy and H

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix A iry function , Comm. Math. Phys. 147 (1992) 1--23 http://projecteuclid.org.ezoris.lib.hokudai.ac.jp/euclid.cmp/1104250524

work page arXiv 1992

[10] [10]

Kimura and A

T. Kimura and A. Zahabi, Universal edge scaling in random partitions , http://dx.doi.org/10.1007/s11005-021-01389-y Lett. Math. Phys. 111 (2021) 48 , http://arxiv.org/abs/2012.06424 arXiv:2012.06424 [cond-mat.stat-mech]

work page doi:10.1007/s11005-021-01389-y 2021

[11] [11]

, Unitary matrix models and random partitions: Universality and multi-criticality , http://dx.doi.org/10.1007/JHEP07(2021)100 JHEP 07 (2021) 100 , http://arxiv.org/abs/2105.00509 arXiv:2105.00509 [hep-th]

work page doi:10.1007/jhep07(2021)100 2021

[12] [12]

, Universal cusp scaling in random partitions, http://dx.doi.org/10.1007/s11005-024-01771-6 Lett. Math. Phys. 114 (2024) Paper No. 27, 56 , http://arxiv.org/abs/2208.07288 arXiv:2208.07288 [math-ph]

work page doi:10.1007/s11005-024-01771-6 2024

[13] [13]

Multicritical edge statistics for the momenta of fermions in non-harmonic traps

P. Le Doussal, S. N. Majumdar, and G. Schehr, Multicritical edge statistics for the momenta of fermions in nonharmonic traps, http://dx.doi.org/10.1103/PhysRevLett.121.030603 Phys. Rev. Lett. 121 (2018) 030603 , http://arxiv.org/abs/1802.06436v1 arXiv:1802.06436v1 [cond-mat.stat-mech]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.121.030603 2018

[14] [14]

B. F. Logan and L. A. Shepp, A variational problem for random Y oung tableaux , http://dx.doi.org/10.1016/0001-8708(77)90030-5 Adv. Math. 26 (1977) 206--222

work page doi:10.1016/0001-8708(77)90030-5 1977

[15] [15]

Matsumoto, Correlation functions of the shifted S chur measure , http://dx.doi.org/10.2969/jmsj/1158241925 J

S. Matsumoto, Correlation functions of the shifted S chur measure , http://dx.doi.org/10.2969/jmsj/1158241925 J. Math. Soc. Japan 57 (2005) 619--637 , http://arxiv.org/abs/math/0312373v2 arXiv:math/0312373v2 [math.CO]

work page doi:10.2969/jmsj/1158241925 2005

[16] [16]

Infinite wedge and random partitions

A. Okounkov, Infinite wedge and random partitions, http://dx.doi.org/10.1007/PL00001398 Selecta Math. 7 (2001) 57--81 , http://arxiv.org/abs/math/9907127 arXiv:math/9907127 [math.RT]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/pl00001398 2001

[17] [17]

C. A. Tracy and H. Widom, Level-spacing distributions and the A iry kernel , http://dx.doi.org/10.1007/BF02100489 Comm. Math. Phys. 159 (1994) 151--174 , http://arxiv.org/abs/hep-th/9211141 arXiv:hep-th/9211141

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02100489 1994

[18] [18]

, A limit theorem for shifted S chur measures , http://dx.doi.org/10.1215/S0012-7094-04-12316-4 Duke Math. J. 123 (2004) 171--208 , http://arxiv.org/abs/math/0210255 arXiv:math/0210255 [math.PR]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1215/s0012-7094-04-12316-4 2004

[19] [19]

A. M. Vershik and S. V. Kerov, Asymptotic behavior of the P lancherel measure of the symmetric group and the limit form of Y oung tableaux , Dokl. Akad. Nauk SSSR 233 (1977) 1024--1027 https://www.mathnet.ru/eng/dan40430

work page 1977

[20] [20]

Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, vol

W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, vol. Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965

work page 1965

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BKP hierarchy and Pfaffian point process

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2018.12.028 2018