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 $d\ge 2$) 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 Hölder 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.

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