We study paths of time-length @math of a continuous-time random walk on @math 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 @math ; 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 @math followed by @math . 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 @math , and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.
As noted above, Berestycki and Yadin @cite_25 (apparently prompted by questions from I. Benjamini) studied a related model of an interacting random walk. There are two notable differences between their and our setting: First, their interaction includes the internal components of the boundary and, second, it is given by the number of on the inner boundary. For this case they showed that the path is confined (with different type of control in @math and @math ) on the spatial scale The exponent is strictly less than @math in all spatial dimensions @math ; the walk is thus squeezed'' by the interaction relative to its typical (diffusive) scaling.
@cite_25
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We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.