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

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 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.