84 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
Superdiffusivity for Brownian Motion in a Poissonian Potential with Long Range Correlation I: Lower Bound on the Volume Exponent
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
We consider the following oriented percolation model of : we equip with the edge set
, and we say that
each edge is open with probability where is a fixed
non-negative compactly supported function on with and is the percolation parameter.
Let denote the percolation threshold ans the number of open
oriented-paths of length starting from the origin, and study the growth of
when percolation occurs. We prove that for if and the function
is sufficiently spread-out, then there exists a second threshold
such that decays exponentially fast for
and does not so when . The result should
extend to the nearest neighbor-model for high-dimension, and for the spread-out
model when . 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
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 the number of self-avoiding paths of length 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 , , 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 , almost surely, 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
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
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