1,057 research outputs found
Weak disorder for low dimensional polymers: The model of stable laws
In this paper, we consider directed polymers in random environment with long
range jumps in discrete space and time. We extend to this case some techniques,
results and classifications known in the usual short range case. However, some
properties are drastically different when the underlying random walk belongs to
the domain of attraction of an \a-stable law. For instance, we construct
natural examples of directed polymers in random environment which experience
weak disorder in low dimension
Directed Polymers in Random Environment are Diffusive at Weak Disorder
In this paper, we consider directed polymers in random environment with
discrete space and time. For transverse dimension at least equal to 3, we prove
that diffusivity holds for the path in the full weak disorder region, i.e.,
where the partition function differs from its annealed value only by a
non-vanishing factor. Deep inside this region, we also show that the quenched
averaged energy has fluctuations of order 1. In complete generality (arbitrary
dimension and temperature), we prove monotonicity of the phase diagram in the
temperature
Shape and local growth for multidimensional branching random walks in random environment
We study branching random walks in random environment on the -dimensional
square lattice, . In this model, the environment has finite range
dependence, and the population size cannot decrease. We prove limit theorems
(laws of large numbers) for the set of lattice sites which are visited up to a
large time as well as for the local size of the population. The limiting shape
of this set is compact and convex, though the local size is given by a concave
growth exponent. Also, we obtain the law of large numbers for the logarithm of
the total number of particles in the process.Comment: 38 pages, 2 figures; to appear in ALE
Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards
We consider a random walk in a stationary ergodic environment in ,
with unbounded jumps. In addition to uniform ellipticity and a bound on the
tails of the possible jumps, we assume a condition of strong transience to the
right which implies that there are no "traps". We prove the law of large
numbers with positive speed, as well as the ergodicity of the environment seen
from the particle. Then, we consider Knudsen stochastic billiard with a drift
in a random tube in , , which serves as environment.
The tube is infinite in the first direction, and is a stationary and ergodic
process indexed by the first coordinate. A particle is moving in straight line
inside the tube, and has random bounces upon hitting the boundary, according to
the following modification of the cosine reflection law: the jumps in the
positive direction are always accepted while the jumps in the negative
direction may be rejected. Using the results for the random walk in random
environment together with an appropriate coupling, we deduce the law of large
numbers for the stochastic billiard with a drift.Comment: 37 pages, 1 figure; to appear in Annales de l'Institut Henri
Poincar\'e (B) Probabilit\'es et Statistique
Finite-size corrections to the speed of a branching-selection process
We consider a particle system studied by E. Brunet and B. Derrida, which
evolves according to a branching mechanism with selection of the fittest
keeping the population size fixed and equal to . The particles remain
grouped and move like a travelling front driven by a random noise with a
deterministic speed. Because of its mean-field structure, the model can be
further analysed as . We focus on the case where the noise lies
in the max-domain of attraction of the Weibull extreme value distribution and
show that under mild conditions the correction to the speed has universal
features depending on the tail probabilities
Rate of convergence for polymers in a weak disorder
We consider directed polymers in random environment on the lattice Z d at
small inverse temperature and dimension d 3. Then, the normalized
partition function W n is a regular martingale with limit W. We prove that n
(d--2)/4 (W n -- W)/W n converges in distribution to a Gaussian law. Both the
polynomial rate of convergence and the scaling with the martingale W n are
different from those for polymers on trees
The vacant set of two-dimensional critical random interlacement is infinite
For the model of two-dimensional random interlacements in the critical regime
(i.e., ), we prove that the vacant set is a.s.\ infinite, thus
solving an open problem from arXiv:1502.03470. Also, we prove that the entrance
measure of simple random walk on annular domains has certain regularity
properties; this result is useful when dealing with soft local times for
excursion processes.Comment: 38 pages, 3 figures; to appear in The Annals of Probabilit
- …