16 research outputs found
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
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
New bounds for the free energy of directed polymers in dimension 1+1 and 1+2
We study the free energy of the directed polymer in random environment in
dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and
Vargas concerning very strong disorder by giving sharp estimates on the free
energy at high temperature. In dimension 2, we prove that very strong disorder
holds at all temperatures, thus solving a long standing conjecture in the
field.Comment: 31 pages, 4 figures, final version, accepted for publication in
Communications in Mathematical Physic
Comments on the Influence of Disorder for Pinning Model in Correlated Gaussian Environment
We study the random pinning model, in the case of a Gaussian environment
presenting power-law decaying correlations, of exponent decay a>0. We comment
on the annealed (i.e. averaged over disorder) model, which is far from being
trivial, and we discuss the influence of disorder on the critical properties of
the system. We show that the annealed critical exponent \nu^{ann} is the same
as the homogeneous one \nu^{pur}, provided that correlations are decaying fast
enough (a>2). If correlations are summable (a>1), we also show that the
disordered phase transition is at least of order 2, showing disorder relevance
if \nu^{pur}<2. If correlations are not summable (a<1), we show that the phase
transition disappears.Comment: 23 pages, 1 figure Modifications in v2 (outside minor typos):
Assumption 1 on correlations has been simplified for more clarity; Theorem 4
has been improved to a more general underlying renewal distribution; Remark
2.1 added, on the assumption on the correlations in the summable cas
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
Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: upper bound on the volume exponent
This paper continues a study on trajectories of Brownian Motion in a field of
soft trap whose radius distribution is unbounded. We show here for both
point-to-point and point-to-plane model the volume exponent (the exponent
associated to transversal fluctuation of the trajectories) is strictly less
than one 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 we get the exact value of the volume
exponent.Comment: 28 page 4 figures, to appear in AIH
Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: upper bound on the volume exponent
This paper continues a study on trajectories of Brownian Motion in a field of
soft trap whose radius distribution is unbounded. We show here for both
point-to-point and point-to-plane model the volume exponent (the exponent
associated to transversal fluctuation of the trajectories) is strictly less
than one 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 we get the exact value of the volume
exponent.Comment: 28 page 4 figures, to appear in AIH