Conditioned random walks and interaction-driven condensation

TL;DR

This paper studies a constrained random walk model, revealing how increasing area constraints induce a condensation phenomenon where a significant portion of the area concentrates in a single excursion, explained via large deviation theory.

Contribution

It introduces a new framework for understanding interaction-driven condensation in constrained random walks using large deviation principles.

Findings

Condensation occurs when the total area constraint is increased.

A finite fraction of the area concentrates in a single excursion during condensation.

The phenomena generalizes previous constraint-induced condensation cases.

Abstract

We consider a discrete-time continuous-space random walk under the constraints that the number of returns to the origin (local time) and the total area under the walk are fixed. We first compute the joint probability of an excursion having area $a$ and returning to the origin for the first time after time $\tau$. We then show how condensation occurs when the total area constraint is increased: an excursion containing a finite fraction of the area emerges. Finally we show how the phenomena generalises previously studied cases of condensation induced by several constraints and how it is related to interaction-driven condensation which allows us to explain the phenomenon in the framework of large deviation theory.