Conditional exponential directed last passage percolation under a one-point upper large deviation event

TL;DR

This paper investigates the behavior of exponential directed last passage percolation under a rare event where the passage time to a specific site is unusually large, revealing new conditional laws and fluctuation limits.

Contribution

It introduces a detailed analysis of the last passage time field conditioned on a one-point upper large deviation event, providing new insights into atypical fluctuations.

Findings

Conditional law of large numbers established

Limiting fluctuations computed in specific regimes

Explicit multi-point distribution analysis used

Abstract

Under typical scaling, the last passage time field of the directed last passage percolation model with exponential site distributions converges to the KPZ fixed point. In this paper, we consider an atypical scenario in which the last passage time to a specific site is unusually large, and we explore how the last passage time field changes under this one-point upper large deviation event. We prove a conditional law of large numbers and compute the limiting fluctuations in certain regimes. Our proofs rely on an analysis of explicit multi-point distributions.