Exact cube-root fluctuations in an area-constrained random walk model

Abstract

This article is devoted to the study of the behaviour of a (1+1)-dimensional model of random walk conditioned to enclose an area of order $N^2$. Such a conditioning enforces a globally concave trajectory. We study the local deviations of the walk from its convex hull. To this end, we introduce two quantities -- the mean facet length $\mathsf{MeanFL}$ and the mean local roughness $\mathsf{MeanLR}$ -- measuring the typical longitudinal and transversal fluctuations around the boundary of the convex hull of the random walk. Our main result is that $\mathsf{MeanFL}$ is of order $N^{2/3}$ and $\mathsf{MeanLR}$ is of order $N^{1/3}$. Moreover, following the strategy of Hammond (Ann. Prob., 2012), we identify the polylogarithmic corrections in the scaling of the maximal facet length and of the maximal local roughness, showing that the former one scales as $N^{2/3}(\log N)^{1/3}$, while the latter scales as $N^{1/3}(\log N)^{2/3}$. The object of study is intended to be a toy model for the interface of a two-dimensional statistical mechanics model (such as the Ising model) in the phase separation regime -- we discuss this issue at the end of this work.