Quenched Invariance Principle for a class of random conductance models with long-range jumps
We study random walks on \({\mathbb Z}^d\) among random conductances \(\{C_{xy}: x,y\in{\mathbb Z}^d\}\) that permit jumps of arbitrary length. Apart from joint ergodicity with respect to shifts, we assume only that the nearest-neighbor conductances are uniformly positive and that \(\sum_{x\in{\mathbb Z}^d} C_{0x}|x|^2\) is integrable.
Our focus is on the Quenched Invariance Principle (QIP) which we establish in all \(d\ge3\) by a combination of corrector methods and heat-kernel technology. We also show that our class contains examples where the corrector is not sublinear everywhere and yet the QIP holds. Thus, although the recent work of Andres, Slowik and Deuschel can be extended to long-range models, it cannot cover all cases for which the QIP is conjectured to hold. Notwithstanding, a combination of their methods with ours proves the QIP for random walks on long-range percolation graphs with exponents larger than \(d+2\) in all \(d\ge2\), provided all nearest-neighbor edges are present.
This is an ongoing joint work with Marek Biskup (UCLA).