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Supercritical self-avoiding walks are space-filling

Hugo Duminil-Copin, Gady Kozma, Ariel Yadin (2014)

Annales de l'I.H.P. Probabilités et statistiques

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory γ between two points on the boundary of a finite subdomain of ℤ d is proportional to μ - length ( γ ) . When μ is supercritical (i.e. μ l t ; μ c where μ c is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

We study trajectories of d -dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii ν has power-law decay and prove that superdiffusivity...

Superdiffusivity for brownian motion in a poissonian potential with long range correlation II: Upper bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) ξ is strictly less than 1 and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and...

Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation

Clément Rau (2007)

Bulletin de la Société Mathématique de France

On s’intéresse à une marche aléatoire simple sur un amas infini issu d’un processus de percolation surcritique sur les arêtes de ℤ d ( d ≥ 2 ) de loi Q . On montre que la transformée de Laplace du nombre de points visités au temps n , noté N n , a un comportement similaire au cas où la marche évolue dans ℤ d . Plus précisément, on établit que pour tout 0 < α < 1 , il existe des constantes C i , C s > 0 telles que pour presque toute réalisation de la percolation telle que l’origine appartienne à l’amas infini et pour n assez grand, e - C i n d / ( d + 2 ) ≤ 𝔼 0 ω ( α N n ) ≤ e - C s n d / ( d + 2 ) . Le...

Survival probability approach to the relaxation of a macroscopic system in the defect-diffusion framework

Paulina Hetman (2004)

Applicationes Mathematicae

The main objective of this paper is to present a new probabilistic model underlying the universal relaxation laws observed in many fields of science where we associate the survival probability of the system's state with the defect-diffusion framework. Our approach is based on the notion of the continuous-time random walk. To derive the properties of the survival probability of a system we explore the limit theorems concerning either the summation or the extremes: maxima and minima. The forms of...

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