259 research outputs found
Organized versus self-organized criticality in the abelian sandpile model
We define stabilizability of an infinite volume height configuration and of a
probability measure on height configurations. We show that for high enough
densities, a probability measure cannot be stabilized. We also show that in
some sense the thermodynamic limit of the uniform measures on the recurrent
configurations of the abelian sandpile model (ASM) is a maximal element of the
set of stabilizable measures. In that sense the self-organized critical
behavior of the ASM can be understood in terms of an ordinary transition
between stabilizable and non-stabilizableComment: 17 pages, appeared in Markov Processes and Related Fields 200
The restriction of the Ising model to a layer
We discuss the status of recent Gibbsian descriptions of the restriction
(projection) of the Ising phases to a layer. We concentrate on the projection
of the two-dimensional low temperature Ising phases for which we prove a
variational principle.Comment: 38 page
Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures
We consider Ising-spin systems starting from an initial Gibbs measure
and evolving under a spin-flip dynamics towards a reversible Gibbs measure
. Both and are assumed to have a finite-range
interaction. We study the Gibbsian character of the measure at time
and show the following: (1) For all and , is Gibbs
for small . (2) If both and have a high or infinite temperature,
then is Gibbs for all . (3) If has a low non-zero
temperature and a zero magnetic field and has a high or infinite
temperature, then is Gibbs for small and non-Gibbs for large
. (4) If has a low non-zero temperature and a non-zero magnetic field
and has a high or infinite temperature, then is Gibbs for
small , non-Gibbs for intermediate , and Gibbs for large . The regime
where has a low or zero temperature and is not small remains open.
This regime presumably allows for many different scenarios
Duality and exact correlations for a model of heat conduction
We study a model of heat conduction with stochastic diffusion of energy. We
obtain a dual particle process which describes the evolution of all the
correlation functions. An exact expression for the covariance of the energy
exhibits long-range correlations in the presence of a current. We discuss the
formal connection of this model with the simple symmetric exclusion process.Comment: 19 page
Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction
We consider three one-dimensional continuous-time Markov processes on a
lattice, each of which models the conduction of heat: the family of Brownian
Energy Processes with parameter , a Generalized Brownian Energy Process, and
the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these
three processes is a parabolic equation, the linear heat equation in the case
of the BEP and the KMP, and a nonlinear heat equation for the GBEP().
We prove the hydrodynamic limit rigorously for the BEP, and give a formal
derivation for the GBEP().
We then formally derive the pathwise large-deviation rate functional for the
empirical measure of the three processes. These rate functionals imply
gradient-flow structures for the limiting linear and nonlinear heat equations.
We contrast these gradient-flow structures with those for processes describing
the diffusion of mass, most importantly the class of Wasserstein gradient-flow
systems. The linear and nonlinear heat-equation gradient-flow structures are
each driven by entropy terms of the form ; they involve dissipation
or mobility terms of order for the linear heat equation, and a
nonlinear function of for the nonlinear heat equation.Comment: 29 page
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Large deviation principle at fixed time in Glauber evolutions
Abstract: We consider the evolution of an asymptotically decoupled probability measure ν on Ising spin configurations under a Glauber dynamics. We prove that for any t > 0, ν t is asymptotically decoupled and hence satisfies a large deviation principle with the relative entropy density as rate function
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