Weak Convergence for Reversible Random Walks in a Random Environment

Quenched invariance principles for walks on clusters of percolation or among random conductances

“…which is proved by Kozlov [19], for arbitrary d ≥ 1, when r ω is close enough to a constant, and by Boivin [5], for general r ω , when d = 2, see also some recent related work of Caputo-Ioffe [9]. We describe here this method because in addition to leading to a proof of (0.12), it might apply in a suitable form to the walk on the infinite cluster of percolation and offer a way of removing the restriction d ≥ 4 in (0.8).…”

“…[19], see also [2] when the starting point is smoothed out by randomization, and for general r ω (·, ·), when d = 1, 2, see [5]. A quenched central limit theorem for the position of the walk at a single time is otherwise shown for general r ω (·, ·) and arbitrary d ≥ 1, in Boivin-Depauw [6].…”

“…We are able to prove a quenched invariance principle in this situation, for general d ≥ 1. For partial results in this direction, we refer to Anshelevich-Khanin-Sinai [2], Boivin [5], Boivin-Depauw [6], Kozlov [19].…”

“…A quenched central limit theorem for the position of the walk at a single time is proved in Boivin-Depauw [6], and otherwise (0.12) is only known when d ≤ 2, cf. Boivin [5], or when r ω (·, ·) is close enough to a deterministic random walk, cf. Kozlov [19], and also Anshelevich-Khanin-Sinai [2], where in addition the starting point of the walk is randomized.…”

Quenched invariance principle for random walks among random degenerate conductances

“…Estimate (57) implies the following L ∞ -estimate on χ jχ j L ∞ (B(n)) sup z∈B(m) B(⌊ n m ⌋z,⌊ n m ⌋)) ω(∇Φ j ) 2 B(⌊ n m ⌋z,2⌊ n m ⌋)) + χ j L 1 (B(⌊ n m ⌋z,2⌊ n m ⌋)) B(⌊ n m ⌋z,⌊ n m ⌋)) ω(∇Φ j ) 2 B(⌊ n m ⌋z,2⌊ n m ⌋)) + m d χ j L 1 (B(2n) + ⌊ n m ⌋. (58)The ergodic theorem in the versions(18) and (56) implies that P-a.s. (B(⌊ n m ⌋z,⌊ n m ⌋)) ω(∇Φ j ) 2 ), (59) and the L 1 -sublinearity of χ j (15), we obtain lim sup…”