26,451 research outputs found
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) \u3e 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)T(f)
Statistical mechanics on isoradial graphs
Isoradial graphs are a natural generalization of regular graphs which give,
for many models of statistical mechanics, the right framework for studying
models at criticality. In this survey paper, we first explain how isoradial
graphs naturally arise in two approaches used by physicists: transfer matrices
and conformal field theory. This leads us to the fact that isoradial graphs
provide a natural setting for discrete complex analysis, to which we dedicate
one section. Then, we give an overview of explicit results obtained for
different models of statistical mechanics defined on such graphs: the critical
dimer model when the underlying graph is bipartite, the 2-dimensional critical
Ising model, random walk and spanning trees and the q-state Potts model.Comment: 22 page
Twistor theory on a finite graph
We show how the description of a shear-free ray congruence in Minkowski space
as an evolving family of semi-conformal mappings can naturally be formulated on
a finite graph. For this, we introduce the notion of holomorphic function on a
graph. On a regular coloured graph of degree three, we recover the space-time
picture. In the spirit of twistor theory, where a light ray is the more
fundamental object from which space-time points should be derived, the line
graph, whose points are the edges of the original graph, should be considered
as the basic object. The Penrose twistor correspondence is discussed in this
context
Graphs and networks theory
This chapter discusses graphs and networks theory
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
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