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Upper tail large deviations for Brownian motions with one-sided collisions

Thomas Weiss

TL;DR

The paper derives explicit upper-tail large deviation principles for Brownian motions with one-sided collisions under three fundamental initial conditions in the KPZ class. It leverages determinantal representations and steepest-descent analysis of carefully scaled kernels to obtain explicit rate functions $r^ extrm{packed}(a)$, $r^ extrm{flat}(a)$, and $r^ extrm{stat}(a)$, with the periodic case uniquely involving Lambert $W$ via $(z)=L_0(z e^z)$. The packed case recovers the known GUE-related rate via a direct kernel limit, while the flat and stationary cases are handled through contour deformations, saddle-point analysis, and analytic continuation (rho-perturbation) to yield the respective rate functions and their asymptotics. The results illuminate the right-tail behavior of the system in KPZ universality, connect to GOE/TW tails in appropriate limits, and provide explicit formulas for use in further probabilistic and integrable-model studies.

Abstract

The system of interacting Brownian motions, where a particle is reflected asymmetrically from its left neighbor, belongs to the KPZ universality class, with multi-point asymptotics having been derived in previous works. In this paper we show upper tail large deviation principles for all three fundamental initial conditions, including explicit calculation of the rate function. For the periodic case the Lambert-W function, which is already present in the Fredholm determinant formula, also appears in the rate function.

Upper tail large deviations for Brownian motions with one-sided collisions

TL;DR

The paper derives explicit upper-tail large deviation principles for Brownian motions with one-sided collisions under three fundamental initial conditions in the KPZ class. It leverages determinantal representations and steepest-descent analysis of carefully scaled kernels to obtain explicit rate functions , , and , with the periodic case uniquely involving Lambert via . The packed case recovers the known GUE-related rate via a direct kernel limit, while the flat and stationary cases are handled through contour deformations, saddle-point analysis, and analytic continuation (rho-perturbation) to yield the respective rate functions and their asymptotics. The results illuminate the right-tail behavior of the system in KPZ universality, connect to GOE/TW tails in appropriate limits, and provide explicit formulas for use in further probabilistic and integrable-model studies.

Abstract

The system of interacting Brownian motions, where a particle is reflected asymmetrically from its left neighbor, belongs to the KPZ universality class, with multi-point asymptotics having been derived in previous works. In this paper we show upper tail large deviation principles for all three fundamental initial conditions, including explicit calculation of the rate function. For the periodic case the Lambert-W function, which is already present in the Fredholm determinant formula, also appears in the rate function.

Paper Structure

This paper contains 5 sections, 9 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Packed initial condition
  4. Periodic initial condition
  5. Poisson initial condition

Key Result

Theorem 1

The system $\{x^\textrm{packed}_n(t),n\in\mathbb{N}\}$ satisfies an upper tail large deviation principle with rate function

Figures (2)

  • Figure 1: The rate function $r^\mathrm{flat}(a)$ (solid line) and its approximating functions according to \ref{['eqFlRateAsy']} at $a\to0$ and $a\to\infty$ (dashed lines).
  • Figure 2: The contour $\gamma$ (dotted line) and its image under $\varphi$ (solid line) for the limiting case $a=0$ (left picture) and some positive $a$ (right picture). The dashed lines separate the ranges of the principal branch $0$ (right) and the branches $1$ (top left) and $-1$ (bottom left) of the Lambert W function.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Remark 1
  • Proposition 2
  • proof : Proof of Theorem \ref{['thmStepRate']}
  • Proposition 3
  • proof : Proof of Theorem \ref{['thmFlatRate']}
  • Lemma 1
  • ...and 6 more