116 research outputs found
Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit
We consider a system consisting of a planar random walk on a square lattice,
submitted to stochastic elementary local deformations. Depending on the
deformation transition rates, and specifically on a parameter which
breaks the symmetry between the left and right orientation, the winding
distribution of the walk is modified, and the system can be in three different
phases: folded, stretched and glassy. An explicit mapping is found, leading to
consider the system as a coupling of two exclusion processes. For all closed or
periodic initial sample paths, a convenient scaling permits to show a
convergence in law (or almost surely on a modified probability space) to a
continuous curve, the equation of which is given by a system of two non linear
stochastic differential equations. The deterministic part of this system is
explicitly analyzed via elliptic functions. In a similar way, by using a formal
fluid limit approach, the dynamics of the system is shown to be equivalent to a
system of two coupled Burgers' equations.Comment: 31 pages, 13 figures. Pages 5,6,8,9,10,12,23 color printed. INRIA
Report 460
Calogero Models for Distinguishable Particles
Motivated by topological bidimensional quantum models for distinguishable
particles, and by Haldane's definition of mutual statistics for different
species of particles, we propose a new class of one-dimensional
Calogero model with coupling constants depending on the labels of the
particles. We solve the groundstate problem, and show how to build some classes
of excited states.Comment: 11 page
Local stability of Belief Propagation algorithm with multiple fixed points
A number of problems in statistical physics and computer science can be
expressed as the computation of marginal probabilities over a Markov random
field. Belief propagation, an iterative message-passing algorithm, computes
exactly such marginals when the underlying graph is a tree. But it has gained
its popularity as an efficient way to approximate them in the more general
case, even if it can exhibits multiple fixed points and is not guaranteed to
converge. In this paper, we express a new sufficient condition for local
stability of a belief propagation fixed point in terms of the graph structure
and the beliefs values at the fixed point. This gives credence to the usual
understanding that Belief Propagation performs better on sparse graphs.Comment: arXiv admin note: substantial text overlap with arXiv:1101.417
The Role of Normalization in the Belief Propagation Algorithm
An important part of problems in statistical physics and computer science can
be expressed as the computation of marginal probabilities over a Markov Random
Field. The belief propagation algorithm, which is an exact procedure to compute
these marginals when the underlying graph is a tree, has gained its popularity
as an efficient way to approximate them in the more general case. In this
paper, we focus on an aspect of the algorithm that did not get that much
attention in the literature, which is the effect of the normalization of the
messages. We show in particular that, for a large class of normalization
strategies, it is possible to focus only on belief convergence. Following this,
we express the necessary and sufficient conditions for local stability of a
fixed point in terms of the graph structure and the beliefs values at the fixed
point. We also explicit some connexion between the normalization constants and
the underlying Bethe Free Energy
About Hydrodynamic Limit of Some Exclusion Processes via Functional Integration
This article considers some classes of models dealing with the dynamics of
discrete curves subjected to stochastic deformations. It turns out that the
problems of interest can be set in terms of interacting exclusion processes,
the ultimate goal being to derive hydrodynamic limits after proper scalings. A
seemingly new method is proposed, which relies on the analysis of specific
partial differential operators, involving variational calculus and functional
integration: indeed, the variables are the values of some functions at given
points, the number of which tends to become infinite, which requires the
construction of \emph{generalized measures}. Starting from a detailed analysis
of the \textsc{asep} system on the torus Z/N/Z, we claim that the arguments a
priori work in higher dimensions (ABC, multi-type exclusion processes, etc),
leading to sytems of coupled partial differential equations of Burgers' type.Comment: Proceedings on CD. ISBN 978-5-901158-15-9; Int. Math. Conf. "50 Years
of IPPI" (2011
A queueing theory approach for a multi-speed exclusion process.
10 pages, 6 figuresInternational audienceWe consider a one-dimensional stochastic reaction-diffusion generalizing the totally asymmetric simple exclusion process, and aiming at describing single lane roads with vehicles that can change speed. To each particle is associated a jump rate, and the particular dynamics that we choose (based on 3-sites patterns) ensures that clusters of occupied sites are of uniform jump rate. When this model is set on a circle or an infinite line, classical arguments allow to map it to a linear network of queues (a zero-range process in theoretical physics parlance) with exponential service times, but with a twist: the service rate remains constant during a busy period, but can change at renewal events. We use the tools of queueing theory to compute the fundamental diagram of the traffic, and show the effects of a condensation mechanism
Pairwise MRF Calibration by Perturbation of the Bethe Reference Point
We investigate different ways of generating approximate solutions to the
pairwise Markov random field (MRF) selection problem. We focus mainly on the
inverse Ising problem, but discuss also the somewhat related inverse Gaussian
problem because both types of MRF are suitable for inference tasks with the
belief propagation algorithm (BP) under certain conditions. Our approach
consists in to take a Bethe mean-field solution obtained with a maximum
spanning tree (MST) of pairwise mutual information, referred to as the
\emph{Bethe reference point}, for further perturbation procedures. We consider
three different ways following this idea: in the first one, we select and
calibrate iteratively the optimal links to be added starting from the Bethe
reference point; the second one is based on the observation that the natural
gradient can be computed analytically at the Bethe point; in the third one,
assuming no local field and using low temperature expansion we develop a dual
loop joint model based on a well chosen fundamental cycle basis. We indeed
identify a subclass of planar models, which we refer to as \emph{Bethe-dual
graph models}, having possibly many loops, but characterized by a singly
connected dual factor graph, for which the partition function and the linear
response can be computed exactly in respectively O(N) and operations,
thanks to a dual weight propagation (DWP) message passing procedure that we set
up. When restricted to this subclass of models, the inverse Ising problem being
convex, becomes tractable at any temperature. Experimental tests on various
datasets with refined or regularization procedures indicate that
these approaches may be competitive and useful alternatives to existing ones.Comment: 54 pages, 8 figure. section 5 and refs added in V
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