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 P. Bella and M. Schäffner. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

arXiv comment36 pages, subsumes salvageable parts of arXiv:1412.0175

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