75 research outputs found
Survival of branching random walks in random environment
We study survival of nearest-neighbour branching random walks in random
environment (BRWRE) on . 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. random matrices.Comment: 17 pages; to appear in Journal of Theoretical Probabilit
Random planar trees and the Jacobian conjecture
We develop a probabilistic approach to the celebrated Jacobian conjecture,
which states that any Keller map (i.e. any polynomial mapping whose Jacobian determinant is a nonzero
constant) has a compositional inverse which is also a polynomial. The Jacobian
conjecture may be formulated in terms of a problem involving labellings of
rooted trees; we give a new probabilistic derivation of this formulation using
multi-type branching processes. Thereafter, we develop a simple and novel
approach to the Jacobian conjecture in terms of a problem about shuffling
subtrees of -Catalan trees, i.e. planar -ary trees. We also show that, if
one can construct a certain Markov chain on large -Catalan trees which
updates its value by randomly shuffling certain nearby subtrees, and in such a
way that the stationary distribution of this chain is uniform, then the
Jacobian conjecture is true. Finally, we show that the subtree shuffling
conjecture is true in a certain asymptotic sense, and thereafter use our
machinery to prove an approximate version of the Jacobian conjecture, stating
that inverses of Keller maps have small power series coefficients for their
high degree terms.Comment: 36 pages, 4 figures. Section 2.5 added, Section 3 expanded, further
minor edit
Survival of near-critical branching Brownian motion
Consider a system of particles performing branching Brownian motion with
negative drift and killed upon hitting zero.
Initially there is one particle at . Kesten showed that the process
survives with positive probability if and only if . Here we are
interested in the asymptotics as \eps\to 0 of the survival probability
. It is proved that if then for all , exists and is a
travelling wave solution of the Fisher-KPP equation. Furthermore, we obtain
sharp asymptotics of the survival probability when and .
The proofs rely on probabilistic methods developed by the authors in a previous
work. This completes earlier work by Harris, Harris and Kyprianou and confirms
predictions made by Derrida and Simon, which were obtained using nonrigorous
PDE methods
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 the particle is typically at a distance of order
from the origin, . We investigate the
probabilities of moderate deviations from this behaviour. Specifically, we are
interested in quenched and annealed probabilities of slowdown (at time , the
particle is at a distance of order from the origin, ), and speedup (at time , the particle is at a distance of order
from the origin, ), for the current location
of the particle and for the hitting times. Also, we study probabilities of
backtracking: at time , the particle is located around , 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
Functional Limit Theorems for Multiparameter Fractional Brownian Motion
We prove a general functional limit theorem for multiparameter fractional
Brownian motion. The functional law of the iterated logarithm, functional
L\'{e}vy's modulus of continuity and many other results are its particular
cases. Applications to approximation theory are discussed.Comment: AMS-LaTeX, 23 page
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
The maximum of the local time of a diffusion process in a drifted Brownian potential
We consider a one-dimensional diffusion process in a
-drifted Brownian potential for . We are interested
in the maximum of its local time, and study its almost sure asymptotic
behaviour, which is proved to be different from the behaviour of the maximum
local time of the transient random walk in random environment. We also obtain
the convergence in law of the maximum local time of under the annealed law
after suitable renormalization when . Moreover, we characterize
all the upper and lower classes for the hitting times of , in the sense of
Paul L\'evy, and provide laws of the iterated logarithm for the diffusion
itself. To this aim, we use annealed technics.Comment: 38 pages, new version, merged with hal-00013040 (arXiv:math/0511053),
with some additional result
Self-intersection local times of random walks: Exponential moments in subcritical dimensions
Fix , not necessarily integer, with . We study the -fold
self-intersection local time of a simple random walk on the lattice up
to time . This is the -norm of the vector of the walker's local times,
. We derive precise logarithmic asymptotics of the expectation of
for scales that are bounded from
above, possibly tending to zero. The speed is identified in terms of mixed
powers of and , and the precise rate is characterized in terms of
a variational formula, which is in close connection to the {\it
Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation
principle for for deviation functions satisfying
t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk
homogeneously squeezes in a -dependent box with diameter of order to produce the required amount of self-intersections. Our main tool is
an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The
final publication is available at springerlink.co
Alternative proof for the localization of Sinai's walk
We give an alternative proof of the localization of Sinai's random walk in
random environment under weaker hypothesis than the ones used by Sinai.
Moreover we give estimates that are stronger than the one of Sinai on the
localization neighborhood and on the probability for the random walk to stay
inside this neighborhood
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