Limit theorems for Random Walk excursion conditioned to have a typical area

TL;DR

This paper establishes a functional central limit theorem for random walk excursions conditioned on a fixed geometric area, advancing understanding of their scaling limits and applications to self-avoiding walk models.

Contribution

It introduces a new functional CLT for conditioned random walk excursions with specific increment properties, extending previous local limit results and aiding analysis of self-avoiding walks.

Findings

Proves a functional CLT for conditioned random walk excursions.

Provides a key tool for analyzing the scaling limit of self-avoiding walks.

Extends results to random walks with Laplace symmetric increments.

Abstract

We derive a functional central limit theorem for the excursion of a random walk conditioned on sweeping a prescribed geometric area. We assume that the increments of the random walk are integer-valued, centered, with a third moment equal to zero and a finite fourth moment. This result complements the work of \citep{DKW13} where local central limit theorems are provided for the geometric area of the excursion of a symmetric random walk with finite second moments. Our result turns out to be a key tool to derive the scaling limit of the \emph{Interacting Partially-Directed Self-Avoiding Walk} at criticality which is the object of a companion paper \citep{CarPet17a}. This requires to derive a reinforced version of our result in the case of a random walk with Laplace symmetric increments.