12,542 research outputs found

### Edge-reinforced random walk on a ladder

We prove that the edge-reinforced random walk on the ladder
${\mathbb{Z}\times\{1,2\}}$ with initial weights $a>3/4$ 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

### 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

### Vortex-boson duality in four space-time dimensions

A continuum version of the vortex-boson duality in (3+1) dimensions is
formulated and its implications studied in the context of a pair Wigner crystal
in underdoped cuprate superconductors. The dual theory to a phase fluctuating
superconductor (or superfluid) is shown to be a theory of bosonic strings
interacting through a Kalb-Ramond rank-2 tensorial gauge field. String
condensation produces Higgs mass for the gauge field and the expected Wigner
crystal emerges as an interesting space-time analog of the Abrikosov lattice.Comment: 4 pages REVTeX; for related work and info visit
http://www.physics.ubc.ca/~fran

### 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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