Invariance principle and local limit theorem for a class of random conductance models with long-range jumps

Abstract

We study continuous time random walks on $\mathbb{Z}^d$ (with $d \geq 2$) among random conductances $\{ \omega(\{x,y\}) : x,y \in \mathbb{Z}^d\}$ that permit jumps of arbitrary length. The law of the random variables $\omega(\{x,y\})$, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Assuming that the first moment of $\sum_{x \in \mathbb{Z}^d} \omega(\{0,x\}) |x|^2$ and the $q$-th moment of $1/\omega(0,x)$ for $x$ neighbouring the origin are finite for some $ q >d/2$, we show a quenched invariance principle and a quenched local limit theorem, where the moment condition is optimal for the latter. We also obtain H\"older regularity estimates for solutions of the heat equation for the associated non-local discrete operator, and deduce that the pointwise spectral dimension equals $d$ almost surely. Our results apply to random walks on long-range percolation graphs with connectivity exponents larger than $d+2$ when all nearest-neighbour edges are present.