Recent progress on the Random Conductance Model

By Marek Tomasz Biskup. 2011

231 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/72e79e8821443598cae01b4d39a8e11d88fd795a

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights (2016, methodology): “In our setting, the generator (∂ t + L ωt ) of the space–time process (t, X t ) is asymmetric and the construction of the corrector as carried out, for instance, in [2, 13] fails, since it is based on a simple projection argument using the symmetry of the generator and an integration by parts.”
A Regularity Theory for Random Elliptic Operators (2014, background): “ons (which is also new). On the one hand, this regularity theory (since it applies to the general ergodic case) is of interest for the study of the random conductance model in probability theory (see [50, 19] for recent surveys). Quenched invariance principles and heat kernel estimates for degenerat”
Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations (2019, background): “g statements with strictly elliptic, unbounded coeﬃcients). In two dimensions, Proposition 2 might be not surprising since an analogous statement in a discrete setting was already proven by Biskup in [7]. For completeness and since the argument in [7] uses the discrete structure we include this case”
Normal approximation for a random elliptic equation (2014, background): “We refer to the recent review paper [3] for many other references on the random conductance model.”
Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “Recently, in [13] Biskup proved the QFCLT for the special case d = 2 under the moment condition (2) with p = q = 1.”