Marek Biskup

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤd with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

Recent progress on the Random Conductance Model

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.

On the scaling of the chemical distance in long-range percolation models

We consider the (unoriented) long-range percolation on Z d in dimensions d ≥ 1, where distinct sites x, y ∈ Z d get connected with probability p xy ∈ [0, 1]. Assuming p xy = |x - y| -s+o(1) as |x - y| → ∞, where s > 0 and |.| is a norm distance on Z d , and supposing that the resulting random graph contains an infinite connected component C∞, we let D(x, y) be the graph distance between x and y measured on C∞. Our main result is that, for s ∈ (d, 2d), D(x, y) = (log|x - y|) Δ+o(1) , x,y ∈ C∞, |x - y| → ∞, where Δ -1 is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x - y| → ∞. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of small-world phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.

Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on

Reflection Positivity and Phase Transitions in Lattice Spin Models

\Z^d

Graph diameter in long-range percolation

,

General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions

d\ge2

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

, driven by a field of bounded random conductances

Rigorous Analysis of Discontinuous Phase Transitions via Mean-Field Bounds

\omega_{xy}\in[0,1]

A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts

. The conductance law is i.i.d. subject to the condition that the probability of