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

Abstract

We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Z d . We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some p ∈ ( 2d d+2 , 2). This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p = 2 to these values.As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator.To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

“…else, (2.13) see [KW15] for further explanation and [GKS07]. We prove that under quite general conditions, this is also true for the random conductance model, see Proposition 2.8 and Corollary 2.9.…”

The Parabolic Anderson Model

“…However, in the case Á < d=2, it turns out in [KönWol15] that the LDP asymptotics follow the formulas of the fixed-box version, which seems to suggest that the random walk fills only a small part of the growing box. Here the local times spread out over the entire large box in a more or less homogeneous way.…”

“…The PAM with RWRC as the underlying random motion has not yet been studied (however, note the work [ErhHolMai15b] on the case of a time-dependent potential with a RWRC with uniformly elliptic random conductances, see Remark 8.7), but important prerequisities have been derived: annealed large-deviation principles (LDPs) for the normalised local times of the RWRC in fixed boxes [KönSalWol12] and in time-depending, growing boxes [KönWol15]; see also the thesis [Wol13]. These LDPs are particularly interesting because of the assumption on the conductances that they are not uniformly elliptic (i.e., not bounded away from zero and from infinity), but can attain arbitrarily small values.…”

“…A continuous analogue to the rate function in Theorem 7.13 is given by [KönWol15] that the minimum of J .c/ 0 is attained for Á > d=2, but not for Á Ä d=2, in which case, in contrast, the discrete version, J .c/ 0 , has a well-defined minimum. In order to properly incorporate the t-dependence of the set B t , like in Sect.…”