Eigenvalue versus perimeter in a shape theorem for self-interacting random walks

Abstract

We study paths of time-length $t$ of a continuous-time random walk on $\mathbb{Z}^{2}$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\beta$; the “energy” is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit $t\to\infty$ followed by $\beta\to\infty$. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in $\mathbb{R}^{2}$, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.