Noam Berger

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.

Emergence of tempered preferential attachment from optimization

We show how preferential attachment can emerge in an optimization framework, resolving a long-standing theoretical controversy. We also show that the preferential attachment model so obtained has two novel features, saturation and viability, which have natural interpretations in the underlying network and lead to a power-law degree distribution with exponential cutoff. Moreover, we consider a generalized version of this preferential attachment model with independent saturation and viability, leading to a broader class of power laws again with exponential cutoff. We present a collection of empirical observations from social, biological, physical, and technological networks, for which such degree distributions give excellent fits. We suggest that, in general, optimization models that give rise to preferential attachment with saturation and viability effects form a good starting point for the analysis of many networks.

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

We consider the nearest-neighbor simple random walk on

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

\Z^d

Degree distribution of the FKP network model

,

Competition-Induced Preferential Attachment

d\ge2

A Quenched Invariance Principle for Certain Ballistic Random Walks in i.i.d. Environments

, driven by a field of bounded random conductances

Degree Distribution of Competition-Induced Preferential Attachment Graphs

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

Long-range percolation mixing time

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

LIMITING VELOCITY OF HIGH-DIMENSIONAL RANDOM WALK IN RANDOM ENVIRONMENT

\omega_{xy}>0