76,429 research outputs found

    Hierarchical pinning model with site disorder: Disorder is marginally relevant

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    We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [6, 9], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [9] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.Comment: 13 pages, 1 figure, final version accepted for publication. to appear in Probability Theory and Related Field

    The Simple Exclusion Process on the Circle has a diffusive Cutoff Window

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    In this paper, we investigate the mixing time of the simple exclusion process on the circle with NN sites, with a number of particle k(N)k(N) tending to infinity, both from the worst initial condition and from a typical initial condition. We show that the worst-case mixing time is asymptotically equivalent to (8π2)1N2logk(8\pi^2)^{-1}N^2\log k, while the cutoff window, is identified to be N2N^2. Starting from a typical condition, we show that there is no cutoff and that the mixing time is of order N2N^2.Comment: 37 pages, 3 Figure

    The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem

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    We consider the stochastic evolution of a 1+1-dimensional interface (or polymer) in presence of a substrate. This stochastic process is a dynamical version of the homogeneous pinning model. We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface. In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by LL in both dimensions and time rescaled by L2L^2 where LL denotes the length of the interface) which we describe: when the interaction with the substrate is such that the system is unpinned at equilibrium, then the scaling limit of the height profile is given by the solution of the heat equation with Dirichlet boundary condition ; when the attraction to the substrate is infinite, the scaling limit is given a free-boundary problem which belongs to the class of Stefan problems with contracting boundary, also referred to as Stefan freezing problems. In addition, we prove the existence and regularity of the solution to this problem until a maximal time, where the boundaries collide.Comment: 42 pages, 4 figures. The paper contains more results than the first version and the title has been change
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