[Paper Review] New edge asymptotics of skew Young diagrams via free boundaries

Dan Betea, Jérémie Bouttier+2 more|arXiv (Cornell University)|Feb 23, 2019

Random Matrices and Applications15 references4 citations

TL;DR

This paper introduces new edge asymptotics for skew Young diagrams under free-boundary Poissonized Plancherel and geometric corner-growth measures, generalizing Tracy–Widom distributions via a KPZ-like 1/3 scaling with logarithmic corrections. Using pfaffian correlation kernels and steepest descent analysis, it derives new limiting distributions that interpolate between classical GUE, GOE, and GSE Tracy–Widom laws and the uniform measure on partitions, with explicit convergence in scaling limits governed by boundary parameters and size parameters.

ABSTRACT

We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.

Motivation & Objective

• To generalize edge scaling limits of skew Young diagrams beyond classical Plancherel and corner-growth measures.
• To derive new limiting distributions that interpolate between Tracy–Widom GUE/GOE/GSE and the uniform measure on partitions.
• To establish a directed LPP model on a 'tie' (reflecting strip) as a physical interpretation of the new measures.
• To extend the KPZ 1/3 fluctuation behavior to skew shapes with free boundaries and logarithmic corrections.
• To provide a rigorous asymptotic analysis of pfaffian point processes arising from skew Schur measures with boundary parameters.

Proposed method

• Introduces two families of measures on skew Young diagrams: upward (↗) and up-down (↗↘) free-boundary Poissonized Plancherel and geometric corner-growth measures.
• Uses Schur functions and specialized generating series to express the measures in terms of semi-standard and standard Young tableaux counts.
• Derives pfaffian correlation kernels for the associated point processes via contour integral representations.
• Applies steepest descent asymptotic analysis to the kernel integrals in the scaling limits M → ∞ and n → ∞.
• Identifies double critical points at z = 1 in the action function, justifying a 1/3 scaling window around the origin.
• Incorporates logarithmic corrections via asymptotic expansions of q-Pochhammer symbols and theta functions, linking them to the observed log terms in the limits.

Experimental results

Research questions

• RQ1 How do edge fluctuations of skew Young diagrams behave under free-boundary Poissonized Plancherel measures with boundary parameters?
• RQ2 What new limiting distributions emerge in the KPZ 1/3 scaling limit with logarithmic corrections for skew shapes?
• RQ3 Can the Tracy–Widom GUE/GOE/GSE laws be generalized to skew diagrams with free boundaries and intermediate measures?
• RQ4 How do the new measures relate to directed last-passage percolation on a reflecting strip (LPP on a tie)?
• RQ5 What role do boundary parameters (ai, bi, u, v) play in shaping the edge asymptotics and limiting distributions?

Key findings

• The edge scaling limit of the first row λ₁ of a skew Young diagram converges to new Tracy–Widom-type distributions F₁;dx(s) and F₂;dx(s) under 1/3 scaling with logarithmic corrections.
• For the Poissonized Plancherel-type measures (M), the limit is F₁;dx(s) with dx = (α₁, α₂; η) for x = aa, (α₁, −; η) for x = ab, and η for x = bb or −.
• For the geometric corner-growth-type measures (fM), the limit is F₁;dx(s) with the same dx classification, under n → ∞ scaling with q = 1 − u².
• The limiting distributions generalize classical Tracy–Widom laws and interpolate between Poissonized Plancherel and uniform measures on partitions.
• The logarithmic corrections in the scaling limits arise from the O(log r) terms in the asymptotic expansion of q-Pochhammer symbols as r → 0.
• The shift 2Dt in the point process can be removed in the limit, as it vanishes in probability under the 1/3 scaling.