33 research outputs found
Inverting Ray-Knight identity
We provide a short proof of the Ray-Knight second generalized Theorem, using
a martingale which can be seen (on the positive quadrant) as the Radon-Nikodym
derivative of the reversed vertex-reinforced jump process measure with respect
to the Markov jump process with the same conductances. Next we show that a
variant of this process provides an inversion of that Ray-Knight identity. We
give a similar result for the Ray-Knight first generalized Theorem.Comment: 18 page
Limit theorems and ergodicity for general bootstrap random walks
Given the increments of a simple symmetric random walk , we
characterize all possible ways of recycling these increments into a simple
symmetric random walk adapted to the filtration of
. We study the long term behavior of a suitably normalized
two-dimensional process . In particular, we provide
necessary and sufficient conditions for the process to converge to a
two-dimensional Brownian motion (possibly degenerate). We also discuss cases in
which the limit is not Gaussian. Finally, we provide a simple necessary and
sufficient condition for the ergodicity of the recycling transformation, thus
generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and
solving the discrete version of the open problem of the ergodicity of the
general L\'evy transformation (see Mansuy and Yor, 2006).Comment: 22 pages, 2 figure
Transient Random Walks in Random Environment on a Galton-Watson Tree
We consider a transient random walk in random environment on a
Galton--Watson tree. Under fairly general assumptions, we give a sharp and
explicit criterion for the asymptotic speed to be positive. As a consequence,
situations with zero speed are revealed to occur. In such cases, we prove that
is of order of magnitude , with . We also
show that the linearly edge reinforced random walk on a regular tree always has
a positive asymptotic speed, which improves a recent result of Collevecchio
\cite{Col06}
Mayer and virial series at low temperature
We analyze the Mayer pressure-activity and virial pressure-density series for
a classical system of particles in continuous configuration space at low
temperature. Particles interact via a finite range potential with an attractive
tail. We propose physical interpretations of the Mayer and virial series'
radius of convergence, valid independently of the question of phase transition:
the Mayer radius corresponds to a fast increase from very small to finite
density, and the virial radius corresponds to a cross-over from monatomic to
polyatomic gas. Our results have consequences for the search of a low density,
low temperature solid-gas phase transition, consistent with the Lee-Yang
theorem for lattice gases and with the continuum Widom-Rowlinson model.Comment: 36 pages, 1 figur
On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model
We consider the coupling from the past implementation of the random-cluster
heat-bath process, and study its random running time, or coupling time. We
focus on hypercubic lattices embedded on tori, in dimensions one to three, with
cluster fugacity at least one. We make a number of conjectures regarding the
asymptotic behaviour of the coupling time, motivated by rigorous results in one
dimension and Monte Carlo simulations in dimensions two and three. Amongst our
findings, we observe that, for generic parameter values, the distribution of
the appropriately standardized coupling time converges to a Gumbel
distribution, and that the standard deviation of the coupling time is
asymptotic to an explicit universal constant multiple of the relaxation time.
Perhaps surprisingly, we observe these results to hold both off criticality,
where the coupling time closely mimics the coupon collector's problem, and also
at the critical point, provided the cluster fugacity is below the value at
which the transition becomes discontinuous. Finally, we consider analogous
questions for the single-spin Ising heat-bath process
General random walk in a random environment defined on Galton–Watson trees
We consider a particle performing a random walk on a Galton–Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours are determined by a random process. We introduce a method for deriving conditions under which the walk is either transient or recurrent. We first suppose that the weights are i.i.d., and re-prove a result of Lyons and Pemantle (Ann. Probab. 20 (1992) 125–136). We then assume a Markovian environment along each line of descent, and finally consider a random walk in a Markovian environment that itself changes the environment. Our approach involves studying the typical behaviour of the walk on fixed lines of descent, which we then show determines the behaviour of the process on the whole tree
BOUNDS ON THE SPEED AND ON REGENERATION TIMES FOR CERTAIN PROCESSES ON REGULAR TREES
We develop a technique that provides a lower bound on the speed of tran- sient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and re- generation time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic [Probab. Theory Related Fields. 122 (2002) 567–592] prove an upper bound of the form b/(b + δ) for the speed on the b-ary tree, where δ is the reinforcement parameter. For δ > 1 we provide a lower bound of the form γ 2b/(b + δ), where γ is the survival probability of an associated branching process