Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Abstract

ABSTRACT. We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on Z 4 ) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability P 2n ω (0, 0) after 2n steps is at most C(ω)n −2 log n, but the best lower bound till now has been C(ω)n −2 . Here we will show that the log n term marks a real phenomenon by constructing an environment, for each sequence λ n → ∞, such thatwith C(ω) > 0 a.s., along a deterministic subsequence of n's. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the d ≥ 5 cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.

Large deviations for the local times of a random walk among random conductances

“…People were interested in deriving laws of large numbers, central limit theorems and invariance principles [SS04, FM06, M08, BP07, BD10, ABDH13] in both the quenched and the annealed setting, under various assumptions on the distribution of the medium. Furthermore, heat kernel estimates [BBHK08] and certain aspects of anomalous behaviour of the walk [BB10] and connections with trapping models [BČ11] were studied. See [B11] for a survey on recent progress on the random conductance model with special emphasis on homogenisation and martingale techniques.…”

Harnack inequalities on weighted graphs and some applications to the random conductance model

“…However, if the conductances are i.i.d. but have fat tails at zero, the QFCLT still holds (see [2]), but due to a trapping phenomenon the heat kernel decay is sub-diffusive, so the transition density does not have enough regularity for a local limit theorem -see [9,10].…”

Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

“…In fact, it is well known that due to a trapping phenomenon under random i.i.d. conductances with sufficiently heavy tails at the zero the subdiffusive heat kernel decay may occur, see [6,7] and cf. [8].…”