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 $\{C_{x,y}\colon x,y\in\mathbb Z^d\}$ 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 $p,q\ge1$ 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 $d\ge2$, 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 $d\ge3$ under the conditions complementary to those of the recent work of P. Bella and M. Sch\"affner. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.