906 research outputs found
Asymptotic behavior of edge-reinforced random walks
In this article, we study linearly edge-reinforced random walk on general
multi-level ladders for large initial edge weights. For infinite ladders, we
show that the process can be represented as a random walk in a random
environment, given by random weights on the edges. The edge weights decay
exponentially in space. The process converges to a stationary process. We
provide asymptotic bounds for the range of the random walker up to a given
time, showing that it localizes much more than an ordinary random walker. The
random environment is described in terms of an infinite-volume Gibbs measure.Comment: Published at http://dx.doi.org/10.1214/009117906000000674 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Edge-reinforced random walk on a ladder
We prove that the edge-reinforced random walk on the ladder
with initial weights is recurrent. The
proof uses a known representation of the edge-reinforced random walk on a
finite piece of the ladder as a random walk in a random environment. This
environment is given by a marginal of a multicomponent Gibbsian process. A
transfer operator technique and entropy estimates from statistical mechanics
are used to analyze this Gibbsian process. Furthermore, we prove spatially
exponentially fast decreasing bounds for normalized local times of the
edge-reinforced random walk on a finite piece of the ladder, uniformly in the
size of the finite piece.Comment: Published at http://dx.doi.org/10.1214/009117905000000396 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bayesian analysis for reversible Markov chains
We introduce a natural conjugate prior for the transition matrix of a
reversible Markov chain. This allows estimation and testing. The prior arises
from random walk with reinforcement in the same way the Dirichlet prior arises
from P\'{o}lya's urn. We give closed form normalizing constants, a simple
method of simulation from the posterior and a characterization along the lines
of W. E. Johnson's characterization of the Dirichlet prior.Comment: Published at http://dx.doi.org/10.1214/009053606000000290 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spontaneous breaking of rotational symmetry in the presence of defects
We prove a strong form of spontaneous breaking of rotational symmetry for a
simple model of two-dimensional crystals with random defects in thermal
equilibrium at low temperature. The defects consist of isolated missing atoms.Comment: 18 page
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