Dyson Ferrari--Spohn diffusions and ordered walks under area tilts

TL;DR

This paper studies non-colliding random walks with area tilts as models for level lines of 2+1D random surfaces, proving their convergence to Ferrari--Spohn diffusions and characterizing their invariant measures.

Contribution

It establishes the convergence of area-tilted non-colliding walks to Ferrari--Spohn diffusions under general conditions, linking discrete models to continuous diffusions.

Findings

Convergence of non-colliding walks to Ferrari--Spohn diffusions.

Invariant measures described by squares of Slater determinants.

Applicable to a broad class of step distributions and potentials.

Abstract

We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of $2+1$-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari--Spohn diffusions associated with limiting Sturm--Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.