On the scaling limits of weakly asymmetric bridges

TL;DR

This paper studies how a discrete bridge with asymmetric growth behavior scales and behaves in the limit, depending on the asymmetry parameter, revealing different regimes of invariant measures, hydrodynamics, and fluctuations.

Contribution

It classifies the asymptotic behaviors of a discrete bridge model under weak asymmetry, depending on the parameter alpha, providing a comprehensive scaling limit analysis.

Findings

Different regimes of invariant measures identified

Hydrodynamic limits characterized for each alpha regime

Fluctuation behaviors classified based on asymmetry strength

Abstract

We consider a discrete bridge from $(0,0)$ to $(2N,0)$ evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order $N^{-\alpha}$ with $\alpha \in (0,\infty)$. We provide a classification of the asymptotic behaviours - invariant measure, hydrodynamic limit and fluctuations - of this model according to the value of the parameter $\alpha$.