176 research outputs found
A combinatorial approach to jumping particles II: general boundary conditions
International audienceWe consider a model of particles jumping on a row, the TASEP. From the point of view of combinatorics a remarkable feauture of this Markov chain is that Catalan numbers are involved in several entries of its stationary distribution. In a companion paper, we gave a combinatorial interpretaion and a simple proof of these observations in the simplest case where the particles enter, jump and exit at the same rate. In this paper we show how to deal with general rates
Duality relations for the ASEP conditioned on a low current
We consider the asymmetric simple exclusion process (ASEP) on a finite
lattice with periodic boundary conditions, conditioned to carry an atypically
low current. For an infinite discrete set of currents, parametrized by the
driving strength , , we prove duality relations which arise from
the quantum algebra symmetry of the generator of the
process with reflecting boundary conditions. Using these duality relations we
prove on microscopic level a travelling-wave property of the conditioned
process for a family of shock-antishock measures for particles: If the
initial measure is a member of this family with microscopic shocks at
positions , then the measure at any time of the process
with driving strength is a convex combination of such measures with
shocks at positions . which can be expressed in terms of
-particle transition probabilities of the conditioned ASEP with driving
strength .Comment: 26 page
Metastability in the dilute Ising model
Consider Glauber dynamics for the Ising model on the hypercubic lattice with
a positive magnetic field. Starting from the minus configuration, the system
initially settles into a metastable state with negative magnetization. Slowly
the system relaxes to a stable state with positive magnetization. Schonmann and
Shlosman showed that in the two dimensional case the relaxation time is a
simple function of the energy required to create a critical Wulff droplet.
The dilute Ising model is obtained from the regular Ising model by deleting a
fraction of the edges of the underlying graph. In this paper we show that even
an arbitrarily small dilution can dramatically reduce the relaxation time. This
is because of a catalyst effect---rare regions of high dilution speed up the
transition from minus phase to plus phase.Comment: 49 page
Conditioned stochastic particle systems and integrable quantum spin systems
We consider from a microscopic perspective large deviation properties of
several stochastic interacting particle systems, using their mapping to
integrable quantum spin systems. A brief review of recent work is given and
several new results are presented: (i) For the general disordered symmectric
exclusion process (SEP) on some finite lattice conditioned on no jumps into
some absorbing sublattice and with initial Bernoulli product measure with
density we prove that the probability of no absorption event
up to microscopic time can be expressed in terms of the generating function
for the particle number of a SEP with particle injection and empty initial
lattice. Specifically, for the symmetric simple exclusion process on conditioned on no jumps into the origin we obtain the explicit first and
second order expansion in of and also to first order in
the optimal microscopic density profile under this conditioning. For the
disordered ASEP on the finite torus conditioned on a very large current we show
that the effective dynamics that optimally realizes this rare event does not
depend on the disorder, except for the time scale. For annihilating and
coalescing random walkers we obtain the generating function of the number of
annihilated particles up to time , which turns out to exhibit some universal
features.Comment: 25 page
Kinetic exchange opinion model: solution in the single parameter map limit
We study a recently proposed kinetic exchange opinion model (Lallouache et.
al., Phys. Rev E 82:056112, 2010) in the limit of a single parameter map.
Although it does not include the essentially complex behavior of the multiagent
version, it provides us with the insight regarding the choice of order
parameter for the system as well as some of its other dynamical properties. We
also study the generalized two- parameter version of the model, and provide the
exact phase diagram. The universal behavior along this phase boundary in terms
of the suitably defined order parameter is seen.Comment: 14 pages, 9 figure
Tropically convex constraint satisfaction
A semilinear relation S is max-closed if it is preserved by taking the
componentwise maximum. The constraint satisfaction problem for max-closed
semilinear constraints is at least as hard as determining the winner in Mean
Payoff Games, a notorious problem of open computational complexity. Mean Payoff
Games are known to be in the intersection of NP and co-NP, which is not known
for max-closed semilinear constraints. Semilinear relations that are max-closed
and additionally closed under translations have been called tropically convex
in the literature. One of our main results is a new duality for open tropically
convex relations, which puts the CSP for tropically convex semilinaer
constraints in general into NP intersected co-NP. This extends the
corresponding complexity result for scheduling under and-or precedence
constraints, or equivalently the max-atoms problem. To this end, we present a
characterization of max-closed semilinear relations in terms of syntactically
restricted first-order logic, and another characterization in terms of a finite
set of relations L that allow primitive positive definitions of all other
relations in the class. We also present a subclass of max-closed constraints
where the CSP is in P; this class generalizes the class of max-closed
constraints over finite domains, and the feasibility problem for max-closed
linear inequalities. Finally, we show that the class of max-closed semilinear
constraints is maximal in the sense that as soon as a single relation that is
not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure
Formulas and Asymptotics for the Asymmetric Simple Exclusion Process
This is an expanded version of a series of lectures delivered by the second
author in June, 2009. It describes the results of three of the authors' papers
on ASEP, from the derivation of exact formulas for configuration probabilities,
through Fredholm determinant representation, to asymptotics for ASEP with step
initial condition establishing KPZ universality. Although complete proofs are
in general not given, at least the main elements of them are.Comment: 25 pages. Version 2 corrects an error in Section II.
The level set method for the two-sided eigenproblem
We consider the max-plus analogue of the eigenproblem for matrix pencils
Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible
values of lambda), which is a finite union of intervals, can be computed in
pseudo-polynomial number of operations, by a (pseudo-polynomial) number of
calls to an oracle that computes the value of a mean payoff game. The proof
relies on the introduction of a spectral function, which we interpret in terms
of the least Chebyshev distance between Ax and lambda Bx. The spectrum is
obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we
explain relation to mean-payoff games and discrete event systems, and show
that the reconstruction of spectrum is pseudopolynomia
Individualization as driving force of clustering phenomena in humans
One of the most intriguing dynamics in biological systems is the emergence of
clustering, the self-organization into separated agglomerations of individuals.
Several theories have been developed to explain clustering in, for instance,
multi-cellular organisms, ant colonies, bee hives, flocks of birds, schools of
fish, and animal herds. A persistent puzzle, however, is clustering of opinions
in human populations. The puzzle is particularly pressing if opinions vary
continuously, such as the degree to which citizens are in favor of or against a
vaccination program. Existing opinion formation models suggest that
"monoculture" is unavoidable in the long run, unless subsets of the population
are perfectly separated from each other. Yet, social diversity is a robust
empirical phenomenon, although perfect separation is hardly possible in an
increasingly connected world. Considering randomness did not overcome the
theoretical shortcomings so far. Small perturbations of individual opinions
trigger social influence cascades that inevitably lead to monoculture, while
larger noise disrupts opinion clusters and results in rampant individualism
without any social structure. Our solution of the puzzle builds on recent
empirical research, combining the integrative tendencies of social influence
with the disintegrative effects of individualization. A key element of the new
computational model is an adaptive kind of noise. We conduct simulation
experiments to demonstrate that with this kind of noise, a third phase besides
individualism and monoculture becomes possible, characterized by the formation
of metastable clusters with diversity between and consensus within clusters.
When clusters are small, individualization tendencies are too weak to prohibit
a fusion of clusters. When clusters grow too large, however, individualization
increases in strength, which promotes their splitting.Comment: 12 pages, 4 figure
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