Recent progress on the Random Conductance Model

Abstract

Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.

80 pages, version published in Prob. Surveys

References in corpus (14)

• Random walks on supercritical percolation clusters
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• Localization and delocalization of random interfaces
• Invariance principle for the random conductance model with unbounded conductances
• Central limit theorem for stationary linear processes
• Extremes of the discrete two-dimensional Gaussian free field
• Anomalous heat-kernel decay for random walk among bounded random conductances
• Disorder, entropy and harmonic functions
• Universality for SLE(4)
• Law of the Iterated Logarithm for the random walk on the infinite percolation cluster
• Quantitative version of the Kipnis-Varadhan theorem and Monte Carlo approximation of homogenized coefficients
• On symmetric random walks with random conductances on ${\mathbb{Z}}^d$
• A quantitative central limit theorem for the random walk among random conductances
• A conditional quenched CLT for random walks among random conductances on ${\mathbb{Z}}^d$

Cited by in corpus (72)

• Elliptic regularity and quantitative homogenization on percolation clusters
• Invariance principle for the random conductance model in a degenerate ergodic environment
• Quenched invariance principle for random walks with time-dependent ergodic degenerate weights
• Limit theory for random walks in degenerate time-dependent random environments
• Estimation of the infinitesimal generator by square-root approximation
• Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights
• Percolation since Saint-Flour
• Dimers and Circle patterns
• Quenched invariance principles for random walks and elliptic diffusions in random media with boundary
• Invariance Principle for the Random Conductance Model with dynamic bounded Conductances
• A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
• Stochastic homogenization of nonconvex discrete energies with degenerate growth
• Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters
• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights
• Hydrodynamics for the partial exclusion process in random environment
• Extrema of the two-dimensional Discrete Gaussian Free Field
• Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla \varphi$ Interface Model
• Green kernel asymptotics for two-dimensional random walks under random conductances
• Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I
• Parabolic Anderson model in a dynamic random environment: random conductances
• Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field
• Random Conductance Models with Stable-like Jumps: Quenched Invariance Principle
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• Random walks in random environments without ellipticity
• Large fluctuations and transport properties of the Lévy-Lorentz gas
• Quenched tail estimate for the random walk in random scenery and in random layered conductance
• Miller-Abrahams random resistor network, Mott random walk and 2-scale homogenization
• Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part II
• An invariance principle for one-dimensional random walks among dynamical random conductances
• Quenched tail estimate for the random walk in random scenery and in random layered conductance II
• Quenched Invariance Principle for a class of random conductance models with long-range jumps
• Invariance Principle for the one-dimensional dynamic Random Conductance Model under Moment Conditions
• Dynamical Freezing in a Spin Glass System with Logarithmic Correlations
• Laws of the Iterated Logarithm for random walks on Random Conductance Models
• Trapping in the random conductance model
• Random walks with unbounded jumps among random conductances I: Uniform quenched CLT
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Annealed invariance principle for random walks on random graphs generated by point processes in ${\mathbb{R}}^d$
• A conditional quenched CLT for random walks among random conductances on ${\mathbb{Z}}^d$
• An invariance principle for one-dimensional random walks in degenerate dynamical random environments
• The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off
• On the speed of Random Walks among Random Conductances
• Random walks among time increasing conductances: heat kernel estimates
• Homogenization of symmetric stable-like processes in stationary ergodic medium
• Stochastic homogenization on randomly perforated domains
• A zero-one law for recurrence and transience of frog processes
• Decay of semigroup for an infinite interacting particle system on continuum configuration spaces
• Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities
• Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium
• Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous
• From quenched invariance principle to semigroup convergence with applications to exclusion processes
• Möbius invariant Y-systems (cluster structures) for Miquel dynamics
• Quantitative homogenization in a balanced random environment
• Rate functions for random walks on Random conuctance Models and related topics
• Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances
• A parameter identification problem in stochastic homogenization
• Quenched invariance principle for a long-range random walk with unbounded conductances
• Large deviations for the local times of a random walk among random conductances in a growing box
• Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment: ${\mathcal{H}}_{-1}$ Suffices
• Recurrence or transience of random walks on random graphs generated by point processes in ${\mathbb{R}}^d$
• A quantitative central limit theorem for the effective conductance on the discrete torus
• Symmetric exclusion as a model of non-elliptic dynamical random conductances
• Coupling between Brownian motion and random walks on the infinite percolation cluster
• Optimal convergence rates in stochastic homogenization in a balanced random environment
• Phase transition of disordered random networks on quasi-transitive graphs
• A central limit theorem for stochastic heat equations in random environment
• Stochastic homogenization of dynamical discrete optimal transport
• Homogenization theory of random walks among deterministic conductances
• Scaling limit of the directional conductivity of random resistor networks on simple point processes
• Dynamical random walk on the integers with a drift
• Quasistatic Droplet percolation