83 research outputs found

    Hierarchical pinning model with site disorder: Disorder is marginally relevant

    Full text link
    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

    Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent

    Get PDF
    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 can be associated to 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 has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by W\"uhtrich: the superdiffusivity phenomenon is enhanced by the presence of correlation.Comment: 28 pages, 3 figures, Title changed, some proof simplified, to appear in AIH

    Existence of an intermediate phase for oriented percolation

    Full text link
    We consider the following oriented percolation model of N×Zd\mathbb {N} \times \mathbb{Z}^d: we equip N×Zd\mathbb {N}\times \mathbb{Z}^d with the edge set {[(n,x),(n+1,y)]nN,x,yZd}\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}, and we say that each edge is open with probability pf(yx)p f(y-x) where f(yx)f(y-x) is a fixed non-negative compactly supported function on Zd\mathbb{Z}^d with zZdf(z)=1\sum_{z\in \mathbb{Z}^d} f(z)=1 and p[0,inff1]p\in [0,\inf f^{-1}] is the percolation parameter. Let pcp_c denote the percolation threshold ans ZNZ_N the number of open oriented-paths of length NN starting from the origin, and study the growth of ZNZ_N when percolation occurs. We prove that for if d5d\ge 5 and the function ff is sufficiently spread-out, then there exists a second threshold pc(2)>pcp_c^{(2)}>p_c such that ZN/pNZ_N/p^N decays exponentially fast for p(pc,pc(2))p\in(p_c,p_c^{(2)}) and does not so when p>pc(2)p> p_c^{(2)}. The result should extend to the nearest neighbor-model for high-dimension, and for the spread-out model when d=3,4d=3,4. It is known that this phenomenon does not occur in dimension 1 and 2.Comment: 16 pages, 2 figures, further typos corrected, enlarged intro and bibliograph

    Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster

    Full text link
    In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on \bbZ^d. More precisely, we count ZNZ_N the number of self-avoiding paths of length NN on the infinite cluster, starting from the origin (that we condition to be in the cluster). We are interested in estimating the upper growth rate of ZNZ_N, lim supNZN1/N\limsup_{N\to \infty} Z_N^{1/N}, that we call the connective constant of the dilute lattice. After proving that this connective constant is a.s.\ non-random, we focus on the two-dimensional case and show that for every percolation parameter p(1/2,1)p\in (1/2,1), almost surely, ZNZ_N grows exponentially slower than its expected value. In other word we prove that \limsup_{N\to \infty} (Z_N)^{1/N} <\lim_{N\to \infty} \bbE[Z_N]^{1/N} where expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walk on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on specifics of percolation on \bbZ^2, so that our result can be extended to a large family of two dimensional models including general self-avoiding walk in random environment.Comment: 25 pages. Version accepted for publication in PTR

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

    Full text link
    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
    corecore