22,226 research outputs found
Combinatorics of lattice paths with and without spikes
We derive a series of results on random walks on a d-dimensional hypercubic
lattice (lattice paths). We introduce the notions of terse and simple paths
corresponding to the path having no backtracking parts (spikes). These paths
label equivalence classes which allow a rearrangement of the sum over paths.
The basic combinatorial quantities of this construction are given. These
formulas are useful when performing strong coupling (hopping parameter)
expansions of lattice models. Some applications are described.Comment: Latex. 25 page
Cardy's Formula for Certain Models of the Bond-Triangular Type
We introduce and study a family of 2D percolation systems which are based on
the bond percolation model of the triangular lattice. The system under study
has local correlations, however, bonds separated by a few lattice spacings act
independently of one another. By avoiding explicit use of microscopic paths, it
is first established that the model possesses the typical attributes which are
indicative of critical behavior in 2D percolation problems. Subsequently, the
so called Cardy-Carleson functions are demonstrated to satisfy, in the
continuum limit, Cardy's formula for crossing probabilities. This extends the
results of S. Smirnov to a non-trivial class of critical 2D percolation
systems.Comment: 49 pages, 7 figure
Asymptotics of correlations in the Ising model: a brief survey
We present a brief survey of rigorous results on the asymptotic behavior of
correlations between two local functions as the distance between their support
diverges, concentrating on the Ising model on with finite-range
ferromagnetic interactions.Comment: Revised version: minor improvements, updated bibliography. Correction
of a (minor but annoying) typo. To appear in Panoramas et Synth\`ese
Random percolation as a gauge theory
Three-dimensional bond or site percolation theory on a lattice can be
interpreted as a gauge theory in which the Wilson loops are viewed as counters
of topological linking with random clusters. Beyond the percolation threshold
large Wilson loops decay with an area law and show the universal shape effects
due to flux tube quantum fluctuations like in ordinary confining gauge
theories. Wilson loop correlators define a non-trivial spectrum of physical
states of increasing mass and spin, like the glueballs of ordinary gauge
theory. The crumbling of the percolating cluster when the length of one
periodic direction decreases below a critical threshold accounts for the finite
temperature deconfinement, which belongs to 2-D percolation universality class.Comment: 20 pages, 14 figure
The Correlation Functions of the XXZ Heisenberg Chain for Zero or Infinite Anisotropy and Random Walks of Vicious Walkers
The XXZ Heisenberg chain is considered for two specific limits of the
anisotropy parameter: \Dl\to 0 and \Dl\to -\infty. The corresponding wave
functions are expressed by means of the symmetric Schur functions. Certain
expectation values and thermal correlation functions of the ferromagnetic
string operators are calculated over the base of N-particle Bethe states. The
thermal correlator of the ferromagnetic string is expressed through the
generating function of the lattice paths of random walks of vicious walkers. A
relationship between the expectation values obtained and the generating
functions of strict plane partitions in a box is discussed. Asymptotic estimate
of the thermal correlator of the ferromagnetic string is obtained in the limit
of zero temperature. It is shown that its amplitude is related to the number of
plane partitions.Comment: 22 pages, 1 figure, LaTe
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