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,y: x, y∈ 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 ∑x∈ZdC0,x|x|2 and q-th moment of 1 / 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 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.