Quenched invariance principle for a class of random conductance models with long-range jumps

Abstract

We study random walks on \({\mathbb {Z}}^d\) (with \(d\ge 2\)) among stationary ergodic random conductances \(\{C_{x,y}:x,y\in {\mathbb {Z}}^d\}\) that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p-th moment of \(\sum _{x\in {\mathbb {Z}}^d}C_{0,x}|x|^2\) and q-th moment of \(1/C_{0,x}\) for x neighboring the origin are finite for some \(p,q\ge 1\) with \(p^{-1}+q^{-1}<2/d\). In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2d in all \(d\ge 2\), provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between \(d+2\) and 2d, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in \(d\ge 3\) under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.