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.
The appearance of the Dirichlet eigenvalue has to do with the large-deviations cost of keeping the walk confined to a given spatial region. The associated large-deviations principle (which goes back to Donsker and Varadhan @cite_11 ) underlies a large body of literature on random walks interacting via their local time and or through an underlying random environment; e.g., the study of the parabolic Anderson model (cf K "onig @cite_9 for a recent review), random walk and or Brownian motion among random obstacles (Sznitman @cite_7 ), etc. Two recent papers of Asselah and Shapira @cite_0 @cite_20 are relevant for our context as they develop a detailed large-deviation approach to the size of the boundary of the random walk range in spatial dimensions @math . This expands on the study of moderate deviations for the Brownian sausage'' by van den Berg, Bolthausen and den Hollander @cite_17 .
@cite_7 @cite_9 @cite_17 @cite_0 @cite_20 @cite_11
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Invariance properties of random walks in bounded domains are a topic of growing interest since they contribute to improving our understanding of diffusion in confined geometries. Recently, limited to Pearson random walks with exponentially distributed straight paths, it has been shown that under isotropic uniform incidence, the average length of the trajectories through the domain is independent of the random walk characteristic and depends only on the ratio of the volume's domain over its surface. In this paper, thanks to arguments of integral geometry, we generalize this property to any isotropic bounded stochastic process and we give the conditions of its validity for isotropic unbounded stochastic processes. The analytical form for the traveled distance from the boundary to the first scattering event that ensures the validity of the Cauchy formula is also derived. The generalization of the Cauchy formula is an analytical constraint that thus concerns a very wide range of stochastic processes, from the original Pearson random walk to a Rayleigh distribution of the displacements, covering many situations of physical importance. An n-step Pearson-Gamma random walk in ℝ d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ℝ d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥d 0 and any n≥2 when q is either (q = d 2 - 1 ) (d 0=3) or q=d−1 (d 0=2) (Le Caer in J. Stat. Phys. 140:728–751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type (n≥2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n 2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201–229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q=d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+floor(n 2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d. We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the Swiss cheese picture of Bolthausen, van den Berg and den Hollander [BBH01]. A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.