76,429 research outputs found
Hierarchical pinning model with site disorder: Disorder is marginally relevant
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
In this paper, we investigate the mixing time of the simple exclusion process
on the circle with sites, with a number of particle 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 , while the cutoff window, is identified to be
. Starting from a typical condition, we show that there is no cutoff and
that the mixing time is of order .Comment: 37 pages, 3 Figure
The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem
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 in both dimensions and time rescaled by where
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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