Annals of Applied Probability

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

Marek Biskup and Eviatar B. Procaccia

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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

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

Received: June 2016
Revised: March 2017
First available in Project Euclid: 3 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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