Random walks on supercritical percolation clusters

DOI10.1214/009117904000000748zbMath1067.60101arXivmath/0302004OpenAlexW2045805203MaRDI QIDQ1769506

Martin T. Barlow

Publication date: 21 March 2005

Published in: The Annals of Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0302004

zbMATH Keywords

Harnack inequalitycontinuous time random walkgraphsheat kernel

Mathematics Subject Classification ID

Random graphs (graph-theoretic aspects) (05C80) Percolation (82B43) Processes in random environments (60K37) Heat and other parabolic equation methods for PDEs on manifolds (58J35)

Related Items (95)

On symmetric random walks with random conductances on \(\mathbb Z^d\) ⋮ Harnack inequalities on weighted graphs and some applications to the random conductance model ⋮ Quenched central limit theorem for random walks in doubly stochastic random environment ⋮ An elliptic Harnack inequality for difference equations with random balanced coefficients ⋮ Gaussian bounds and collisions of variable speed random walks on lattices with power law conductances ⋮ Biased random walk on supercritical percolation: anomalous fluctuations in the ballistic regime ⋮ Spectral dimension of simple random walk on a long-range percolation cluster ⋮ Un théorème limite central local en environnement aléatoire stationnaire de conductances sur Z ⋮ On chemical distances and shape theorems in percolation models with long-range correlations ⋮ Gaussian bounds and parabolic Harnack inequality on locally irregular graphs ⋮ Quenched invariance principle for simple random walk on clusters in correlated percolation models ⋮ Quenched invariance principle for simple random walk on percolation clusters ⋮ Invariance principle for the random conductance model ⋮ Supercritical loop percolation on \(\mathbb{Z}^d\) for \(d \geq 3\) ⋮ Invariance principle for Mott variable range hopping and other walks on point processes ⋮ Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights ⋮ Disorder, entropy and harmonic functions ⋮ Equality of critical parameters for percolation of Gaussian free field level sets ⋮ Simple random walk on long range percolation clusters. I: Heat kernel bounds ⋮ Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments ⋮ Off-diagonal heat kernel estimates for symmetric diffusions in a degenerate ergodic environment ⋮ Mini-workshop: New horizons in motions in random media. Abstracts from the mini-workshop held February 26 -- March 4, 2023 ⋮ Quantitative Russo-Seymour-Welsh for random walk on random graphs and decorrelation of uniform spanning trees ⋮ Separation profile, isoperimetry, growth and compression ⋮ Limit theory for random walks in degenerate time-dependent random environments ⋮ Uniformity of the uncovered set of random walk and cutoff for lamplighter chains ⋮ Two random walks on the open cluster of \(\mathbb Z^{2}\) meet infinitely often ⋮ Transience and anchored isoperimetric dimension of supercritical percolation clusters ⋮ Trapping in the random conductance model ⋮ Existence of the harmonic measure for random walks on graphs and in random environments ⋮ Sharp phase transition for Gaussian percolation in all dimensions ⋮ Harmonic deformation of Delaunay triangulations ⋮ Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model ⋮ The maximum of log‐correlated Gaussian fields in random environment ⋮ Green kernel asymptotics for two-dimensional random walks under random conductances ⋮ Recent progress on the random conductance model ⋮ Unnamed Item ⋮ Simple random walk on long-range percolation clusters. II: Scaling limits ⋮ Convergence to fractional kinetics for random walks associated with unbounded conductances ⋮ Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions ⋮ Discrete fractal dimensions of the ranges of random walks in \(\mathbb Z^d\) associate with random conductances ⋮ Rate function of large deviation for a class of nonhomogeneous Markov chains on supercritical percolation network ⋮ An interlacing technique for spectra of random walks and its application to finite percolation clusters ⋮ Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation ⋮ A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment ⋮ Quenched invariance principle for simple random walk on discrete point processes ⋮ Random walk on the incipient infinite cluster for oriented percolation in high dimensions ⋮ First-order expansion of homogenized coefficients under Bernoulli perturbations ⋮ Strongly correlated random interacting processes. Abstracts from the workshop held January 28 -- February 3, 2018 ⋮ Quenched invariance principle for random walks in balanced random environment ⋮ Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs ⋮ Multi-scale Lipschitz percolation of increasing events for Poisson random walks ⋮ Volume growth and stochastic completeness of graphs ⋮ Effective resistances for supercritical percolation clusters in boxes ⋮ Optimal corrector estimates on percolation cluster ⋮ Random conductance models with stable-like jumps: quenched invariance principle ⋮ A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment ⋮ Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights ⋮ Existence of a non-averaging regime for the self-avoiding walk on a high-dimensional infinite percolation cluster ⋮ Random walk on the high-dimensional IIC ⋮ Heat kernel estimates for strongly recurrent random walk on random media ⋮ Quenched invariance principles for random walks with random conductances ⋮ Quasistatic droplets in randomly perforated domains ⋮ Surface order large deviations for 2D FK-percolation and Potts models ⋮ Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster ⋮ Heat-kernel estimates for random walk among random conductances with heavy tail ⋮ Invariance principle for the random conductance model with unbounded conductances ⋮ The discrete integral maximum principle and its applications ⋮ Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice ⋮ Fractal dimension of discrete sets and percolation ⋮ Quenched invariance principle for the Knudsen stochastic billiard in a random tube ⋮ Random walks among time increasing conductances: heat kernel estimates ⋮ Quenched invariance principles for random walks on percolation clusters ⋮ Random walks in random conductances: decoupling and spread of infection ⋮ Heat kernel estimates for random walks on some kinds of one-dimensional continuum percolation clusters ⋮ Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution ⋮ Percolation Hamiltonians ⋮ A note on exit time for anchored isoperimetry ⋮ Anomalous heat-kernel decay for random walk among bounded random conductances ⋮ Asymptotic behaviors for percolation clusters with uncorrelated weights ⋮ Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters ⋮ Quenched invariance principle for a class of random conductance models with long-range jumps ⋮ Some results for range of random walk on graph with spectral dimension two ⋮ Directed polymers on infinite graphs ⋮ Lower Gaussian heat kernel bounds for the random conductance model in a degenerate ergodic environment ⋮ Recurrence or transience of random walks on random graphs generated by point processes in \(\mathbb{R}^d\) ⋮ Disconnection and entropic repulsion for the harmonic crystal with random conductances ⋮ Harry Kesten's work in probability theory ⋮ The Alexander-Orbach conjecture holds in high dimensions ⋮ Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities ⋮ Random walks on discrete point processes ⋮ Anomalous Behavior of Random Walks on Disordered Media ⋮ Quenched invariance principles for random walks and elliptic diffusions in random media with boundary ⋮ Invariance principle for the random conductance model in a degenerate ergodic environment ⋮ Branching random walks and multi-type contact-processes on the percolation cluster of \(\mathbb{Z}^{d}\)

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The construction of Brownian motion on the Sierpinski carpet
• An invariance principle for reversible Markov processes. Applications to random motions in random environments
• Opérateurs uniformément sous-elliptiques sur les groupes de Lie. (Uniformly sub-elliptic operators on Lie groups)
• The incipient infinite cluster in two-dimensional percolation
• Subdiffusive behavior of random walk on a random cluster
• Upper bounds for symmetric Markov transition functions
• A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash
• Transition densities for Brownian motion on the Sierpinski carpet
• Isoperimetric inequalities and transient random walks on graphs
• Dirichlet forms on fractals: Poincaré constant and resistance
• Parabolic Harnack inequality and estimates of Markov chains on graphs
• Random walk on the infinite cluster of the percolation model
• Domination by product measures
• Markov chain approximations to symmetric diffusions
• On the mixing time of a simple random walk on the super critical percolation cluster
• Return probabilities of a simple random walk on percolation clusters
• The Poincaré inequality for vector fields satisfying Hörmander's condition
• Isoperimetry and heat kernel decay on percolation clusters.
• Quenched invariance principles for walks on clusters of percolation or among random conduc\-tances
• Surface order large deviations for Ising, Potts and percolation models
• Surface order large deviations for high-density percolation
• Surface order large deviations for 2D FK-percolation and Potts models
• On the chemical distance for supercritical Bernoulli percolation
• THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS
• ON ARONSON'S UPPER BOUNDS FOR HEAT KERNELS
• Continuity of Solutions of Parabolic and Elliptic Equations
• Percolation
• Large Deviations for Heat Kernels on Graphs
• Stability of parabolic Harnack inequalities
• Large deviations for discrete and continuous percolation

This page was built for publication: Random walks on supercritical percolation clusters