Eigenvalue vs 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.

arXiv comment31 pages, 1 figure, see arXiv:1603.03871 for a companion analysis paper, version to appear in Ann. Appl. Probab

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Cited by in corpus (3)

• The random walk penalised by its range in dimensions $d\ge 3$
• Percolation for the Finitary Random interlacements
• The concentration inequality for a discrete height function model perturbed by random potential