41 research outputs found
Brownian motion and Random Walk above Quenched Random Wall
We study the persistence exponent for the first passage time of a random walk
below the trajectory of another random walk. More precisely, let and
be two centered, weakly dependent random walks. We establish that
for a
non-random . In the classical setting, , it is
well-known that . We prove that for any non-trivial one has
and the exponent depends only on
.
Our result holds also in the continuous setting, when and are
independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck
processes. In the latter case the probability decays at exponential rate.Comment: To appear in Ann. Inst. Henri Poincar\'e Probab. Sta
U-statistics of Ornstein-Uhlenbeck branching particle system
We consider a branching particle system consisting of particles moving
according to the Ornstein-Uhlenbeck process in \Rd and undergoing a binary,
supercritical branching with a constant rate . This system is known
to fulfil a law of large numbers (under exponential scaling). Recently the
question of the corresponding central limit theorem has been addressed. It
turns out that the normalization and form of the limit in the CLT fall into
three qualitatively different regimes, depending on the relation between the
branching intensity and the parameters of the Orstein-Uhlenbeck process. In the
present paper we extend those results to -statistics of the system proving a
law of large numbers and a central limit theorem.Comment: References update. arXiv admin note: substantial text overlap with
arXiv:1007.171
A note on the discrete Gaussian Free Field with disordered pinning on Z^d, d\geq 2
We study the discrete massless Gaussian Free Field on , , in
the presence of a disordered square-well potential supported on a finite strip
around zero. The disorder is introduced by reward/penalty interaction
coefficients, which are given by i.i.d. random variables. Under minimal
assumptions on the law of the environment, we prove that the quenched free
energy associated to this model exists in , is deterministic, and
strictly smaller than the annealed free energy whenever the latter is strictly
positive.Comment: 17 page
The random interchange process on the hypercube
We prove the occurrence of a phase transition accompanied by the emergence of
cycles of diverging lengths in the random interchange process on the hypercube.Comment: 8 page