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Asymptotic expansion for multiplicative statistics in a Hermitian matrix model connected to the lower tail of the KPZ equation

Carla Mariana da Silva Pinheiro

TL;DR

This work analyzes multiplicative statistics for a unitary ensemble under a growing parameter deformation of the weight, extending prior bounded-deformation results to $x=x_0 n^{\alpha}$ with $\alpha\in(0,2/9)$ and linking the large-$n$ limit to the KPZ lower-tail problem. The authors develop a rigorous Riemann–Hilbert framework for orthogonal polynomials with a deformed weight, build global and local parametrices around the spectral endpoints, and perform small-norm analysis to extract precise asymptotics. A key novelty is the connection between the matrix-model deformation and the KPZ lower tail via the Claeys–Cafasso model, allowing the dominant contributions to originate from a neighborhood of the origin and yielding explicit derivatives and leading terms for the multiplicative statistics. The results establish a precise correspondence between large-$n$ statistics of the deformed random matrix model and KPZ fluctuations, with implications for tail behavior and normalization constants in the associated orthogonal-polynomial ensemble.

Abstract

We explore the multiplicative statistics for a unitary random matrix ensemble with a parameter-dependent deformation inserted in the probability measure. Such deformations had been studied for a bounded or decaying parameter. In this work, we extend the previous results for a growing parameter under a controlled rate, and show that the underlying statistics relate to the lower tail study for the KPZ equation.

Asymptotic expansion for multiplicative statistics in a Hermitian matrix model connected to the lower tail of the KPZ equation

TL;DR

This work analyzes multiplicative statistics for a unitary ensemble under a growing parameter deformation of the weight, extending prior bounded-deformation results to with and linking the large- limit to the KPZ lower-tail problem. The authors develop a rigorous Riemann–Hilbert framework for orthogonal polynomials with a deformed weight, build global and local parametrices around the spectral endpoints, and perform small-norm analysis to extract precise asymptotics. A key novelty is the connection between the matrix-model deformation and the KPZ lower tail via the Claeys–Cafasso model, allowing the dominant contributions to originate from a neighborhood of the origin and yielding explicit derivatives and leading terms for the multiplicative statistics. The results establish a precise correspondence between large- statistics of the deformed random matrix model and KPZ fluctuations, with implications for tail behavior and normalization constants in the associated orthogonal-polynomial ensemble.

Abstract

We explore the multiplicative statistics for a unitary random matrix ensemble with a parameter-dependent deformation inserted in the probability measure. Such deformations had been studied for a bounded or decaying parameter. In this work, we extend the previous results for a growing parameter under a controlled rate, and show that the underlying statistics relate to the lower tail study for the KPZ equation.

Paper Structure

This paper contains 15 sections, 27 theorems, 218 equations, 3 figures.

Table of Contents

  1. Introduction and statement of the results
  2. Outline of the paper
  3. Equilibrium measure and related functions
  4. Model local problem
  5. The Riemann-Hilbert problem for the lower tail of the KPZ equation
  6. A Riemann-Hilbert problem for the local parametrix
  7. Small norm for the error problem
  8. The Riemann-Hilbert approach for Orthogonal Polynomials
  9. Local parametrix around $-a$
  10. Local parametrix around $0$
  11. Small norm for the orthogonal polynomials RHP
  12. Multiplicative statistics
  13. Proof of Proposition \ref{['theo1']}
  14. Proof of Theorem \ref{['theo2']}
  15. Proof of Theorem \ref{['theo3']}

Key Result

Proposition 1.2

Let $Q$ be under Assumption asump2 and let $\alpha$ fall under one of the two cases of Assumption asump1. Take $t_0\in(0,1)$ a real constant and $t \in [t_0, 1/t_0]$. Let $\omega_n(z;x) = {\rm e}^{-nV(z)}\sigma_n(z)$ and set $K_n^Q(\lambda, \mu;x)$ to be the Christoffel–Darboux kernel of orthogonal uniformly in both $x=x_0n^{\alpha}$ and $t \in [t_0, 1/t_0]$, as $n \to \infty$.

Figures (3)

  • Figure 1: Domains for deformation of the contour.
  • Figure 2: Opening of lenses.
  • Figure 3: Contour for the Riemann-Hilbert problem ${\rm R}(z)$.

Theorems & Definitions (35)

  • Proposition 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1: GG21
  • Remark 2.2
  • Proposition 3.1: Proposition 3.5, CC2019
  • Lemma 3.3
  • ...and 25 more