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Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

Leslie Molag, Guilherme L. F. Silva, Lun Zhang

TL;DR

This work analyzes the local statistics at the hard edge for Laguerre-type orthogonal polynomial ensembles under a position-dependent multiplicative deformation, interpreting it as conditional thinning. Using a novel Riemann-Hilbert framework with nonconstant jumps, the authors derive a universal limiting kernel, the conditional thinned Bessel kernel $\mathsf K_\alpha$, expressed via a function $\Phi$ solving a nonlocal integrable system that generalizes Painlevé-type structures. They establish a deformation representation for multiplicative statistics, showing convergence to $\mathsf L_\alpha^{(Bes)}$, and connect the nonlocal equations to a Markovian-type Lax pair; for $m=1$ the nonlocal equation reduces to a local PDE capturing the same integrable structure as recent Bessel-process work by Ruzza. The analysis combines a carefully constructed model RHP, a steepest-descent analysis, and global-to-local parametrices to yield precise kernel and statistic asymptotics, thereby extending hard-edge universality to conditional thinning and clarifying its integrable underpinnings with potential applications to related multiplicative-statistics settings.

Abstract

We study the local statistics of orthogonal polynomial ensembles near a hard edge, subject to a multiplicative deformation of the measure. Probabilistically, this deformation corresponds to a position-dependent conditional thinning of the particles. We prove that, under critical hard edge scaling and for a large class of potentials and deformation symbols, the correlation kernel of the conditional ensemble converges to a universal limit, which we identify as the conditional thinned Bessel point process. We derive an explicit expression for this limiting kernel in terms of the solution to a nonlocal integrable system depending on a parameter. For a special choice of the parameter, this system was recently identified in the study of multiplicative statistics of the Bessel point process. Our results establish that this system governs the full correlation structure of the conditional Bessel point process, extending the classical connection between the standard Bessel kernel and the Painlevé V equation.

Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

TL;DR

This work analyzes the local statistics at the hard edge for Laguerre-type orthogonal polynomial ensembles under a position-dependent multiplicative deformation, interpreting it as conditional thinning. Using a novel Riemann-Hilbert framework with nonconstant jumps, the authors derive a universal limiting kernel, the conditional thinned Bessel kernel , expressed via a function solving a nonlocal integrable system that generalizes Painlevé-type structures. They establish a deformation representation for multiplicative statistics, showing convergence to , and connect the nonlocal equations to a Markovian-type Lax pair; for the nonlocal equation reduces to a local PDE capturing the same integrable structure as recent Bessel-process work by Ruzza. The analysis combines a carefully constructed model RHP, a steepest-descent analysis, and global-to-local parametrices to yield precise kernel and statistic asymptotics, thereby extending hard-edge universality to conditional thinning and clarifying its integrable underpinnings with potential applications to related multiplicative-statistics settings.

Abstract

We study the local statistics of orthogonal polynomial ensembles near a hard edge, subject to a multiplicative deformation of the measure. Probabilistically, this deformation corresponds to a position-dependent conditional thinning of the particles. We prove that, under critical hard edge scaling and for a large class of potentials and deformation symbols, the correlation kernel of the conditional ensemble converges to a universal limit, which we identify as the conditional thinned Bessel point process. We derive an explicit expression for this limiting kernel in terms of the solution to a nonlocal integrable system depending on a parameter. For a special choice of the parameter, this system was recently identified in the study of multiplicative statistics of the Bessel point process. Our results establish that this system governs the full correlation structure of the conditional Bessel point process, extending the classical connection between the standard Bessel kernel and the Painlevé V equation.
Paper Structure (34 sections, 30 theorems, 381 equations, 2 figures)

This paper contains 34 sections, 30 theorems, 381 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. The model RHP
  4. The model problem
  5. Solvability of the model problem
  6. Differential equations for the model problem with some particular data
  7. The model RHP: admissible data and asymptotics
  8. The class of admissible $\mathsf h$
  9. Asymptotic analysis for $\bm \Psi_\infty$ as $\mathsf s\to +\infty$
  10. Asymptotic analysis of $\bm \Psi_\infty$ as $\mathsf u\to +\infty$
  11. Asymptotic analysis of the model RHP with admissible data
  12. Consequences of the asymptotic analysis for the model problem
  13. Convergence of the kernel
  14. Asymptotics for a relevant integral
  15. Proof of main theorems on integrable equations
  16. ...and 19 more sections

Key Result

Theorem 2.3

Fix $\alpha>-1$. For $(\zeta,\mathsf s,\mathsf x)\in \mathbb{R}\times \mathbb{R} \times (0,+\infty)$, the nonlocal differential equation admits a solution $\Phi=\Phi(\zeta\mid \mathsf s,\mathsf x)$ satisfying for each $\mathsf x>0$ fixed, Furthermore, as $\mathsf x\to 0^+$, uniformly for $\zeta\in \mathbb{R}\setminus \{0\}$ and $s\in\mathbb{R}$, where $I_\alpha$ is the modified Bessel function

Figures (2)

  • Figure 1: The contours $\Gamma_{\pm}$, $\Gamma_0$, and the sectors $\mathcal{S}_{\pm}$, $\mathcal{S}_{0}$.
  • Figure 2: The contour $\Gamma_{\bm S}$ for RHP \ref{['rhp:S']}.

Theorems & Definitions (58)

  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Theorem 3.5
  • ...and 48 more