2,698 research outputs found
Asymptotics of the Invariant Measure in Mean Field Models with Jumps
We consider the asymptotics of the invariant measure for the process of the
empirical spatial distribution of coupled Markov chains in the limit of a
large number of chains. Each chain reflects the stochastic evolution of one
particle. The chains are coupled through the dependence of the transition rates
on this spatial distribution of particles in the various states. Our model is a
caricature for medium access interactions in wireless local area networks. It
is also applicable to the study of spread of epidemics in a network. The
limiting process satisfies a deterministic ordinary differential equation
called the McKean-Vlasov equation. When this differential equation has a unique
globally asymptotically stable equilibrium, the spatial distribution
asymptotically concentrates on this equilibrium. More generally, its limit
points are supported on a subset of the -limit sets of the
McKean-Vlasov equation. Using a control-theoretic approach, we examine the
question of large deviations of the invariant measure from this limit.Comment: 58 pages, reorganised to get quickly to the main results on invariant
measure; Stochastic Systems, volume 2, 201
Probability around the Quantum Gravity. Part 1: Pure Planar Gravity
In this paper we study stochastic dynamics which leaves quantum gravity
equilibrium distribution invariant. We start theoretical study of this dynamics
(earlier it was only used for Monte-Carlo simulation). Main new results concern
the existence and properties of local correlation functions in the
thermodynamic limit. The study of dynamics constitutes a third part of the
series of papers where more general class of processes were studied (but it is
self-contained), those processes have some universal significance in
probability and they cover most concrete processes, also they have many
examples in computer science and biology. At the same time the paper can serve
an introduction to quantum gravity for a probabilist: we give a rigorous
exposition of quantum gravity in the planar pure gravity case. Mostly we use
combinatorial techniques, instead of more popular in physics random matrix
models, the central point is the famous exponent.Comment: 40 pages, 11 figure
A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
These notes are based on lectures delivered by the authors at a Langeoog
seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a
mixed audience of mathematicians and theoretical physicists. After a brief
outline of the basic physical concepts of equilibrium and nonequilibrium
states, the one-dimensional simple exclusion process is introduced as a
paradigmatic nonequilibrium interacting particle system. The stationary measure
on the ring is derived and the idea of the hydrodynamic limit is sketched. We
then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and
explain the associated universality conjecture for surface fluctuations in
growth models. This is followed by a detailed exposition of a seminal paper of
Johansson that relates the current fluctuations of the totally asymmetric
simple exclusion process (TASEP) to the Tracy-Widom distribution of random
matrix theory. The implications of this result are discussed within the
framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
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