## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 1 (2018), 340-377.

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

Marek Biskup and Eviatar B. Procaccia

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 1 (2018), 340-377.

**Dates**

Received: June 2016

Revised: March 2017

First available in Project Euclid: 3 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1520046090

**Digital Object Identifier**

doi:10.1214/17-AAP1307

**Mathematical Reviews number (MathSciNet)**

MR3770879

**Zentralblatt MATH identifier**

06873686

**Subjects**

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]

**Keywords**

Interacting polymer random environment shape theorem first-passage percolation Dirichlet eigenvalue perimeter

#### Citation

Biskup, Marek; Procaccia, Eviatar B. Eigenvalue versus perimeter in a shape theorem for self-interacting random walks. Ann. Appl. Probab. 28 (2018), no. 1, 340--377. doi:10.1214/17-AAP1307. https://projecteuclid.org/euclid.aoap/1520046090