112 research outputs found
Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster
We consider i.i.d. random variables {\omega (b):b \in E_d} parameterized by
the family of bonds in Z^d, d>1. The random variable \omega(b) is thought of as
the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming
the probability m of the event {\omega(b)>0} to be supercritical and denoting
by C(\omega) the unique infinite cluster associated to the bonds with positive
conductance, we study the zero range process on C(\omega) with
\omega(b)-proportional probability rate of jumps along bond b. For almost all
realizations of the environment we prove that the hydrodynamic behavior of the
zero range process is governed by the nonlinear heat equation , where the matrix D and the function
\phi are \omega--independent. We do not require any ellipticity condition.Comment: 30 pages, new results (see Appendix A), final versio
Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
We consider a stationary and ergodic random field {\omega(b)} parameterized
by the family of bonds b in Z^d, d>1. The random variable \omega(b) is thought
of as the conductance of bond b and it ranges in a finite interval [0,c_0].
Assuming that the set of bonds with positive conductance has a unique infinite
cluster C, we prove homogenization results for the random walk among random
conductances on C. As a byproduct, applying the general criterion of \cite{F}
leading to the hydrodynamic limit of exclusion processes with bond-dependent
transition rates, for almost all realizations of the environment we prove the
hydrodynamic limit of simple exclusion processes among random conductances on
C. The hydrodynamic equation is given by a heat equation whose diffusion matrix
does not depend on the environment. We do not require any ellipticity
condition. As special case, C can be the infinite cluster of supercritical
Bernoulli bond percolation.Comment: 24 pages. extensions and corrections. new titl
Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models
We consider a family X^{(n)}, n \in \bbN_+, of continuous-time
nearest-neighbor random walks on the one dimensional lattice Z. We reduce the
spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions
outside (0,n) to the analogous problem for a suitable generalized second order
differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given
interval. If the measures dm_n weakly converge to some measure dm_*, we prove a
limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the
corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove
the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a
consequence, we establish lower and upper bounds for the asymptotic annealed
eigenvalue counting functions in the case that m is a self--similar stochastic
process. Finally, we apply the above results to investigate the spectral
structure of some classes of subdiffusive random trap and barrier models coming
from one-dimensional physics.Comment: Published on EJP. the Dirichlet-Neumann bracketing has been
corrected. shorter, improved version. 35 page
Mott law as upper bound for a random walk in a random environment
We consider a random walk on the support of an ergodic simple point process
on R^d, d>1, furnished with independent energy marks. The jump rates of the
random walk decay exponentially in the jump length and depend on the energy
marks via a Boltzmann-type factor. This is an effective model for the
phonon-induced hopping of electrons in disordered solids in the regime of
strong Anderson localization. Under mild assumptions on the point process we
prove an upper bound of the asymptotic diffusion matrix of the random walk in
agreement with Mott law. A lower bound in agreement with Mott law was proved in
\cite{FSS}.Comment: 22 pages. Additional results and corrections
Hydrodynamic limit of a disordered lattice gas
We consider a model of lattice gas dynamics in the d-dimensional cubic
lattice in the presence of disorder. If the particle interaction is only mutual
exclusion and if the disorder field is given by i.i.d. bounded random
variables, we prove the almost sure existence of the hydrodynamical limit in
dimension d>2. The limit equation is a non linear diffusion equation with
diffusion matrix characterized by a variational principle
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