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Edgeworth-type expansion for the one-point distribution of the KPZ fixed point with a large height at a prior location

Ron Nissim, Ruixuan Zhang

TL;DR

This work analyzes the KPZ fixed point with narrow-wedge initial data under the rare event of a large height at an earlier time, establishing an Edgeworth-type expansion for the conditional one-point distribution when the later time satisfies $\tau>\tau'$. The authors derive explicit contour-integral formulas for the joint and conditional distributions, then perform a steepest-descent analysis showing the leading term equals the GUE Tracy–Widom distribution with two lower-order corrections given by $\mathrm{F}'$ and $\mathrm{F}''$, plus an $O(h'^{-3/2})$ remainder. The expansion reveals a KPZ-specific refinement beyond the central limit paradigm and contrasts with the $\tau<\tau'$ Gaussian-type limits, thereby mapping the phase diagram of conditional KPZ behavior. The approach relies on precise control of Airy-type kernels, Vandermonde structures, and iterated residues, contributing sharp asymptotics and a framework for higher-order corrections in integrable KPZ models.

Abstract

We consider the Kardar-Parisi-Zhang (KPZ) fixed point $\mathrm{H}(x,τ)$ with the narrow-wedge initial condition and investigate the distribution of $\mathrm{H}(x,τ)$ conditioned on a large height at an earlier space-time point $\mathrm{H}(x',τ')$. As $\mathrm{H}(x',τ')$ tends to infinity, we prove that the conditional one-point distribution of $\mathrm{H}(x,τ)$ in the regime $τ>τ'$ converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution and that the next two lower-order error terms can be expressed as derivatives of the Tracy-Widom distribution. The lowe order expansion here is analogue to the Edgeworth expansion in the central limit theorem. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in \cite{Liu-Wang22} where they study the finite-dimensional distribution of $\mathrm{H}(x,τ)$ conditioned on a large height at a later space-time point $\mathrm{H}(x',τ')$. They show, with the narrow-wedge initial condition, that the conditional random field $\mathrm{H}(x,τ)$ in the regime $τ<τ'$ converges to the minimum of two independent Brownian bridges modified by linear drifts as $\mathrm{H}(x',τ')$ goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes $τ>τ'$ and $τ<τ'$ when $\mathrm{H}(x',τ')$ goes to infinity.

Edgeworth-type expansion for the one-point distribution of the KPZ fixed point with a large height at a prior location

TL;DR

This work analyzes the KPZ fixed point with narrow-wedge initial data under the rare event of a large height at an earlier time, establishing an Edgeworth-type expansion for the conditional one-point distribution when the later time satisfies . The authors derive explicit contour-integral formulas for the joint and conditional distributions, then perform a steepest-descent analysis showing the leading term equals the GUE Tracy–Widom distribution with two lower-order corrections given by and , plus an remainder. The expansion reveals a KPZ-specific refinement beyond the central limit paradigm and contrasts with the Gaussian-type limits, thereby mapping the phase diagram of conditional KPZ behavior. The approach relies on precise control of Airy-type kernels, Vandermonde structures, and iterated residues, contributing sharp asymptotics and a framework for higher-order corrections in integrable KPZ models.

Abstract

We consider the Kardar-Parisi-Zhang (KPZ) fixed point with the narrow-wedge initial condition and investigate the distribution of conditioned on a large height at an earlier space-time point . As tends to infinity, we prove that the conditional one-point distribution of in the regime converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution and that the next two lower-order error terms can be expressed as derivatives of the Tracy-Widom distribution. The lowe order expansion here is analogue to the Edgeworth expansion in the central limit theorem. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in \cite{Liu-Wang22} where they study the finite-dimensional distribution of conditioned on a large height at a later space-time point . They show, with the narrow-wedge initial condition, that the conditional random field in the regime converges to the minimum of two independent Brownian bridges modified by linear drifts as goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes and when goes to infinity.
Paper Structure (14 sections, 9 theorems, 108 equations, 1 figure)

This paper contains 14 sections, 9 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on the KPZ universality class and the KPZ fixed point.
  3. Main results.
  4. Preliminaries
  5. Explicit formula for the joint distribution.
  6. Explicit formula for the conditional tail distribution.
  7. Proof of Theorem \ref{['thm:main']}.
  8. Proof of Lemma \ref{['mainlm']}
  9. Estimates on the exponents of $\mathrm{f}_\ell\left(\boldsymbol{\xi}^{(\ell)}\right),\mathrm{f}_\ell\left(\boldsymbol{\eta}^{(\ell)}\right)$.
  10. Estimates on determinants.
  11. Estimates on iterated residues.
  12. Appendix
  13. Proof of Lemma \ref{['I1bound']}
  14. Proof of Lemma \ref{['Error Lemma for n_1=1']}

Key Result

Theorem 1.1

Suppose $x,x',\mathrm{h}\in \mathbb{R}$ and $\tau,\tau'>0$ are fixed constants. Then as $\mathrm{h}'\to \infty$, where $\mathrm{F}'(\mathrm{h};x,\tau)$ and $\mathrm{F}"(\mathrm{h};x,\tau)$ stand for the first and second order derivatives of $\mathrm{F}(\mathrm{h};x,\tau)$. Here the conditional probability $\mathbb{P}(A \mid \mathrm{H}(x',\tau')=\mathrm{h}')$ is understood as $\lim_{

Figures (1)

  • Figure 1: Integration of contours in Proposition \ref{['prop1']}.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 10 more