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

We study random walks on Z(d) (with d >= 2) among stationary ergodic random conductances {C-x,C-y : x, y is an element of 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 Sigma(x is an element of Zd) C-0,C-x vertical bar x vertical bar(2) and q-th moment of 1/C-0,C-x for x neighboring the origin are finite for some p, q >= 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 >= 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 >= 3 under the conditions complementary to those of the recent work of Bella and Schaffner (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.