[Paper Review] Multicritical random partitions

Dan Betea, Jérémie Bouttier+1 more|arXiv (Cornell University)|Dec 3, 2020

Random Matrices and Applications18 references4 citations

TL;DR

This paper introduces multicritical Schur measures on integer partitions whose edge fluctuations exhibit a critical exponent of $1/(2n+1)$, differing from the generic $1/3$ found in the Tracy–Widom GUE distribution. Using asymptotic analysis of Fredholm determinants and connections to orthogonal polynomials, the authors show that the distribution of the first part $\lambda_1$ converges to a higher-order analogue of the Tracy–Widom distribution, specifically the $\mathcal{A}_{2n+1}(x,y)$ kernel tied to the Painlev\'e II hierarchy, and establish an exact mapping to Periwal–Shevitz multicritical unitary matrix models.

ABSTRACT

We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the first part asymptotically follows a "higher-order analogue" of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz.

Motivation & Objective

• To construct Schur measures on integer partitions with edge behavior characterized by a critical exponent $1/(2n+1)$, distinct from the generic $1/3$.
• To identify the limiting distribution of $\lambda_1$ as a higher-order analogue of the Tracy–Widom GUE distribution, tied to the Painlev\'e II hierarchy.
• To establish a precise mapping between the proposed Schur measures and the multicritical unitary matrix models of Periwal and Shevitz.
• To compute the limit shapes of the partitions under these measures and analyze their asymptotic behavior.

Proposed method

• The authors define Schur measures via specialized Schur functions with parameters tuned to induce multicritical edge behavior.
• They use the Jacobi–Trudi identity and generating functions to express the Schur functions in terms of power sums and exponential generating series.
• Asymptotic analysis is performed using steepest descent methods on contour integrals representing the kernel $K(k,\ell)$, focusing on the scaling regime $\ell = b\theta + s(\theta d)^{1/(2n+1)}$ as $\theta \to \infty$.
• The kernel $K(k,\ell)$ is shown to converge to the continuous integral operator $\mathcal{A}_{2n+1}(x,y)$, whose Fredholm determinant yields the limiting distribution $F(2n+1;s)$.
• The connection to unitary matrix models is established via the identity $\mathbb{E}_{U\in\mathcal{U}(\ell)}[\exp\operatorname{tr}\tilde{V}(U+U^*))$ being equal to the normalized partition function of the Schur measure.
• Analytical tools such as dominated convergence and tail bounds are used to justify the convergence of discrete Fredholm determinants to their continuous counterparts.

Experimental results

Research questions

• RQ1 How can Schur measures be constructed to exhibit edge fluctuations with a critical exponent $1/(2n+1)$ instead of the generic $1/3$?
• RQ2 What is the limiting distribution of $\lambda_1$ in such measures, and how does it relate to known special functions or hierarchies?
• RQ3 Is there a direct correspondence between these Schur measures and the multicritical unitary matrix models of Periwal and Shevitz?
• RQ4 How do the limit shapes of the partitions behave asymptotically under these measures?
• RQ5 Can the connection between fermionic models with nonharmonic traps and random matrix models be explained combinatorially through Schur measures?

Key findings

• The distribution of $\lambda_1$ in the odd-case multicritical Schur measure converges to the Fredholm determinant $F(2n+1;s) = \det(1 - \mathcal{A}_{2n+1})_{L^2(s,\infty)}$ as $\theta \to \infty$, with scaling $\ell = b\theta + s(\theta d)^{1/(2n+1)}$, where $b = 2^{4n-1}n^{-1}\binom{2n}{n}^{-2}$ and $d = \frac{(2n-1)!!}{(2n-2)!!}$.
• The kernel $K(k,\ell)$ converges to $\mathcal{A}_{2n+1}(x,y)$, which is a $\tau$-function of the $2n+1$-th member of the Painlev\'e II hierarchy.
• The quantities $\mathbb{P}^\mathrm{o}_{n,\theta}(\lambda_1 \leq \ell)$, $\mathbb{P}^\mathrm{o}_{n,\theta}(\lambda'_1 \leq \ell)$, the normalized determinant $\det_{1\leq i,j\leq\ell}[f_{j-i}]/e^{\sum \theta_{2i-1}^2/(2i-1)}$, and the matrix model expectation $\mathbb{E}_{U\in\mathcal{U}(\ell)}[\exp\operatorname{tr}\tilde{V}(U+U^*)]/e^{\sum \theta_{2i-1}^2/(2i-1)}$ are all equal and converge to $F(2n+1;s)$.
• For $n=1$, the model reduces to the poissonized Plancherel measure, and the matrix model side recovers the Gross–Witten model.
• The limit shape of the partition is derived from the asymptotic behavior of the measure, with the edge scaling governed by the $1/(2n+1)$ exponent.
• The work provides a combinatorial explanation for the coincidence between fermionic models with nonharmonic traps and multicritical unitary matrix models noted by Le Doussal, Majumdar, and Schehr.