619 research outputs found

### Quenched central limit theorem for the stochastic heat equation in weak disorder

We continue with the study of the mollified stochastic heat equation in
$d\geq 3$ given by $d u_{\epsilon,t}=\frac 12\Delta u_{\epsilon,t}+ \beta
\epsilon^{(d-2)/2} \,u_{\epsilon,t} \,d B_{\epsilon,t}$ with spatially
smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac
solution describes the partition function of the continuous directed polymer.
In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending
on the value of $\beta>0$ in the limiting object of the smoothened solution
$u_\epsilon$ as the smoothing parameter $\epsilon\to 0$ This partition function
naturally defines a quenched polymer path measure and we prove that as long as
$\beta>0$ stays small enough while $u_\epsilon$ converges to a strictly
positive non-degenerate random variable, the distribution of the diffusively
rescaled Brownian path converges under the aforementioned polymer path measure
to standard Gaussian distribution.Comment: Minor revisio

### Multiple scattering in random mechanical systems and diffusion approximation

This paper is concerned with stochastic processes that model multiple (or
iterated) scattering in classical mechanical systems of billiard type, defined
below. From a given (deterministic) system of billiard type, a random process
with transition probabilities operator P is introduced by assuming that some of
the dynamical variables are random with prescribed probability distributions.
Of particular interest are systems with weak scattering, which are associated
to parametric families of operators P_h, depending on a geometric or mechanical
parameter h, that approaches the identity as h goes to 0. It is shown that (P_h
-I)/h converges for small h to a second order elliptic differential operator L
on compactly supported functions and that the Markov chain process associated
to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L
are selfadjoint (densely) defined on the space L2(H,{\eta}) of
square-integrable functions over the (lower) half-space H in R^m, where {\eta}
is a stationary measure. This measure's density is either (post-collision)
Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes
with infinitesimal generator L respectively correspond to what we call MB
diffusion and (generalized) Legendre diffusion. Concrete examples of simple
mechanical systems are given and illustrated by numerically simulating the
random processes.Comment: 34 pages, 13 figure

### Fluctuations of the front in a stochastic combustion model

We consider an interacting particle system on the one dimensional lattice
$\bf Z$ modeling combustion. The process depends on two integer parameters
$2\le a<M<\infty$. Particles move independently as continuous time simple
symmetric random walks except that 1. When a particle jumps to a site which has
not been previously visited by any particle, it branches into $a$ particles; 2.
When a particle jumps to a site with $M$ particles, it is annihilated. We start
from a configuration where all sites to the left of the origin have been
previously visited and study the law of large numbers and central limit theorem
for $r_t$, the rightmost visited site at time $t$.
The proofs are based on the construction of a renewal structure leading to a
definition of regeneration times for which good tail estimates can be
performed.Comment: 19 page

### The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature

We study a spin system on a large box with both Ising interaction and
Sherrington-Kirpatrick couplings, in the presence of an external field. Our
results are: (i) existence of the pressure in the limit of an infinite box.
When both Ising and Sherrington-Kirpatrick temperatures are high enough, we
prove that: (ii) the value of the pressure is given by a suitable replica
symmetric solution, and (iii) the fluctuations of the pressure are of order of
the inverse of the square of the volume with a normal distribution in the
limit. In this regime, the pressure can be expressed in terms of random field
Ising models

### 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

### Survival of branching random walks in random environment

We study survival of nearest-neighbour branching random walks in random
environment (BRWRE) on ${\mathbb Z}$. A priori there are three different
regimes of survival: global survival, local survival, and strong local
survival. We show that local and strong local survival regimes coincide for
BRWRE and that they can be characterized with the spectral radius of the first
moment matrix of the process. These results are generalizations of the
classification of BRWRE in recurrent and transient regimes. Our main result is
a characterization of global survival that is given in terms of Lyapunov
exponents of an infinite product of i.i.d. $2\times 2$ random matrices.Comment: 17 pages; to appear in Journal of Theoretical Probabilit

### Stretched Polymers in Random Environment

We survey recent results and open questions on the ballistic phase of
stretched polymers in both annealed and quenched random environments.Comment: Dedicated to Erwin Bolthausen on the occasion of his 65th birthda

### Knudsen gas in a finite random tube: transport diffusion and first passage properties

We consider transport diffusion in a stochastic billiard in a random tube
which is elongated in the direction of the first coordinate (the tube axis).
Inside the random tube, which is stationary and ergodic, non-interacting
particles move straight with constant speed. Upon hitting the tube walls, they
are reflected randomly, according to the cosine law: the density of the
outgoing direction is proportional to the cosine of the angle between this
direction and the normal vector. Steady state transport is studied by
introducing an open tube segment as follows: We cut out a large finite segment
of the tube with segment boundaries perpendicular to the tube axis. Particles
which leave this piece through the segment boundaries disappear from the
system. Through stationary injection of particles at one boundary of the
segment a steady state with non-vanishing stationary particle current is
maintained. We prove (i) that in the thermodynamic limit of an infinite open
piece the coarse-grained density profile inside the segment is linear, and (ii)
that the transport diffusion coefficient obtained from the ratio of stationary
current and effective boundary density gradient equals the diffusion
coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law
and equality of transport diffusion and self-diffusion coefficients for quite
generic rough (random) tubes. We also study some properties of the crossing
time and compute the Milne extrapolation length in dependence on the shape of
the random tube.Comment: 51 pages, 3 figure

### On slowdown and speedup of transient random walks in random environment

We consider one-dimensional random walks in random environment which are
transient to the right. Our main interest is in the study of the sub-ballistic
regime, where at time $n$ the particle is typically at a distance of order
$O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the
probabilities of moderate deviations from this behaviour. Specifically, we are
interested in quenched and annealed probabilities of slowdown (at time $n$, the
particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in
(0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order
$n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location
of the particle and for the hitting times. Also, we study probabilities of
backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus
making an unusual excursion to the left. For the slowdown, our results are
valid in the ballistic case as well.Comment: 43 pages, 4 figures; to appear in Probability Theory and Related
Field

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