Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights

Abstract

We consider a stationary and ergodic random field $\{\omega (e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb Z^d$, $d \ge 2$. The random variable $\omega (e)$, taking values in $[0, \infty )$ and satisfying certain moment bounds, is thought of as the conductance of the edge e. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal C_{\infty }(\omega )$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under certain moment conditions. An essential ingredient of our proof is a new anchored relative isoperimetric inequality.

Appendix: Ergodic theorem

In this appendix we provide an extension of the Birkhoff ergodic theorem that generalises the result obtained in [13, Theorem 3]. Consider a probability space $(\Omega , \mathcal F, {{\mathrm{\mathbb {P}}}})$ and a group of measure preserving transformations $\tau _x : \Omega \rightarrow \Omega$, $x \in \mathbb Z^d$ such that $\tau _{x+y}= \tau _x\circ \tau _y$. Further, let $B_1 \mathrel {\mathop :}=\{ x \in \mathbb R^d : |x| \le 1 \}$.

Theorem 4.1

Let $\varphi \in L^{1}({{\mathrm{\mathbb {P}}}})$ and $\varepsilon \in (0, d)$. Then, for ${{\mathrm{\mathbb {P}}}}$-a.e. $\omega$,

(4.1)

where the summation is taken over all $x \in B(0, n) {\setminus } \{0\}$.

Proof

To start with, notice that the ergodic theorem, see [13, Theorem 3], implies that for ${{\mathrm{\mathbb {P}}}}$-a.e. $\omega$

(4.2)

On the other hand, by means of Abel’s summation formula, we have that

where we used that $j^{-(d-\varepsilon )} - (j+1)^{-(d-\varepsilon )} \le (d-\varepsilon ) j^{-(d+1-\varepsilon )}$. From this estimate we deduce that for any $\varphi \in L^{1}({{\mathrm{\mathbb {P}}}})$ and ${{\mathrm{\mathbb {P}}}}$-a.e. $\omega$

(4.3)

where $C \mathrel {\mathop :}=\sup _{n \ge 1} |B(0, n)| / n^d < \infty$. On the other hand, for any $k \ge 1$

(4.4)

Since the last factor on the right-hand side of (4.4) is finite due to (4.3), we conclude that ${{\mathrm{\mathbb {P}}}}$-a.s.

(4.5)

uniformly in n. The assertion follows by combining (4.2) and (4.5). $\square$