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$q$-deformation of random partitions, determinantal structure, and Riemann-Hilbert problem

Taro Kimura

TL;DR

The paper studies $q$-deformations of Plancherel-type measures on partitions, formulating two main models and revealing their determinantal structure. It derives a $q$-Bessel kernel from the Schur measure, establishes Toeplitz-determinant expressions for gap probabilities, and develops a Riemann–Hilbert framework for the associated $q$-orthogonal polynomials, leading to $q$-Painlevé dynamics. In the scaling limit $q\to1$ with $\xi=(1-q)\eta$, the results recover the discrete Bessel kernel and, in bulk and edge regimes, the sine and Airy universality classes, respectively, along with a corresponding limit-shape description. The work connects random partitions under $q$-deformed measures to integrable systems through Lax pairs and Painlevé equations, enriching the interplay between combinatorics, representation theory, and mathematical physics.

Abstract

We study $q$-deformation of probability measures on partitions, i.e., $q$-deformed random partitions. We in particular consider the $q$-Plancherel measure and show a determinantal formula for the correlation function using a $q$-deformation of the discrete Bessel kernel. We also investigate Riemann-Hilbert problems associated with the corresponding orthogonal polynomials and obtain $q$-Painlevé equations from the $q$-difference Lax formalism.

$q$-deformation of random partitions, determinantal structure, and Riemann-Hilbert problem

TL;DR

The paper studies -deformations of Plancherel-type measures on partitions, formulating two main models and revealing their determinantal structure. It derives a -Bessel kernel from the Schur measure, establishes Toeplitz-determinant expressions for gap probabilities, and develops a Riemann–Hilbert framework for the associated -orthogonal polynomials, leading to -Painlevé dynamics. In the scaling limit with , the results recover the discrete Bessel kernel and, in bulk and edge regimes, the sine and Airy universality classes, respectively, along with a corresponding limit-shape description. The work connects random partitions under -deformed measures to integrable systems through Lax pairs and Painlevé equations, enriching the interplay between combinatorics, representation theory, and mathematical physics.

Abstract

We study -deformation of probability measures on partitions, i.e., -deformed random partitions. We in particular consider the -Plancherel measure and show a determinantal formula for the correlation function using a -deformation of the discrete Bessel kernel. We also investigate Riemann-Hilbert problems associated with the corresponding orthogonal polynomials and obtain -Painlevé equations from the -difference Lax formalism.

Paper Structure

This paper contains 14 sections, 27 theorems, 170 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. $q$-Plancherel measure
  3. Reduction from Schur measure
  4. $q$-Bessel kernel
  5. Scaling limit
  6. Gap probability and orthogonal polynomials
  7. Toeplitz determinant formula
  8. Orthogonal polynomials on the unit circle
  9. Relation to $q$-orthogonal polynomials
  10. Riemann--Hilbert problem
  11. Weight function $\mathbb{I}$
  12. Weight function $\check{\mathbb{I}}$
  13. $q$-functions
  14. $q$-Bessel functions

Key Result

Theorem 1.1

The correlation kernel of the squared type $q$-Plancherel measure is given by the $q$-Bessel kernel, where $J_n = J^{(3)}_n(2\xi;q)$ is the Hahn--Exton $q$-Bessel function.

Figures (1)

  • Figure 1: The limit shape functions for the parameter $\xi \in [0,1)$ (step 0.1) for the squared type measure.

Theorems & Definitions (44)

  • Theorem 1.1: Theorem \ref{['thm:q-Bessel_kernel']}
  • Proposition 1.2: Propositions \ref{['prop:gap_prob_unitary_matrix']}, \ref{['prop:Toeplitz_Z']}
  • Theorem 1.3: Theorems \ref{['thm:qPV1']}, \ref{['thm:qPV2']}
  • Proposition 2.1
  • proof
  • Theorem 2.2: Okounkov Okounkov:2001SM
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 34 more