1,721,012 research outputs found
Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit
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Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster
Full text linkWe consider i.i.d. random variables {ω(b): b ∈ Ed} parameterized by the family of bonds in Zd, d ≥ 2. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c0]. Assuming the probability m of the event {ω(b) > 0} to be supercritical and denoting by C (ω) the unique infinite cluster associated to the bonds with positive conductance, we study the zero range process on C (ω) with ω(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 ∂tρ = m∇ · (D∇φ(ρ/m)), where the matrix D and the function φ are ω-independent. As byproduct of the above result and the blocking effect of the finite clusters, we discuss the bulk behavior of the zero range process on Zd with conductance field ω. We do not require any ellipticity condition
The alternating marked point process of h-slopes of the drifted Brownian motion
No full textAbstractWe show that the slopes between h-extrema of the drifted 1D Brownian motion form a stationary alternating marked point process, extending the result of J. Neveu and J. Pitman for the non-drifted case. Our analysis covers the results on the statistics of h-extrema obtained by P. Le Doussal, C. Monthus and D. Fisher via a Renormalization Group analysis and gives a complete description of the slope between h-extrema covering the origin by means of the Palm–Khinchin theory. Moreover, we analyze the behavior of the Brownian motion near its h-extrema
Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization
Full text linkWe consider continuous-time random walks on a random locally finite subset of
with random symmetric jump probability rates. The jump range can
be unbounded. We assume some second--moment conditions and that the above
randomness is left invariant by the action of the group
or . We then add a
site-exclusion interaction, thus making the particle system a simple exclusion
process. We show that, for almost all environments, under diffusive space-time
rescaling the system exhibits a hydrodynamic limit in path space. The
hydrodynamic equation is non-random and governed by the effective homogenized
matrix of the single random walk, which can be degenerate. The above result
covers a very large family of models including e.g. simple exclusion processes
built from random conductance models on and on crystal lattices
(possibly with long conductances), Mott variable range hopping, simple random
walks on Delaunay triangulations, random walks on supercritical percolation
clusters.Comment: 43 pages. Minor corrections and extensions. Extended Section 5 with
further applications. Added new Appendix A with an example of degenerate
nonzero effective homogenized matri
Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models
No full textWe consider a sequence X-(n), n >= 1, 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-mn D-x, with Dirichlet conditions outside a given interval. If the measures dm(n) weakly converge to some measure dm(infinity), we prove a limit theorem for the eigenvalues and eigenfunctions of - D-mn D-x to the corresponding spectral quantities of - D-m infinity 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
Discrete kinetic models for molecular motors: asymptotic velocity and gaussian fluctuations
No full textMott law as upper bound for a random walk in a random environment
No full textWe consider a random walk on the support of an ergodic simple point process on R-d, d >= 2, 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 some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d >= 2 in agreement with Mott law was proved in [8]
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