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Author: Wolff, Tilman × WGL Institute: WIAS ×

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    Large deviations for the local times of a random walk among random conductances
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) König, Wolfgang; Salvi, Michele; Wolff, Tilman
    We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances 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 an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.
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    Precise asymptotics for the parabolic Anderson model with a moving catalyst or trap
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Schnitzler, Adrian; Wolff, Tilman
    We consider the solution ucolon [0,infty) timesmathbbZ^drightarrow [0,infty) to the parabolic Anderson model, where the potential is given by (t,x)mapstogammadelta_Y_tleft(xright) with Y a simple symmetric random walk on mathbbZ^d. Depending on the parameter gammain[-infty,infty), the potential is interpreted as a randomly moving catalyst or trap. In the trap case, i.e., gamma<0, we look at the annealed time asymptotics in terms of the first moment of u. Given a localized initial condition, we derive the asymptotic rate of decay to zero in dimensions 1 and 2 up to equivalence and characterize the limit in dimensions 3 and higher in terms of the Green's function of a random walk. For a homogeneous initial condition we give a characterisation of the limit in dimension 1 and show that the moments remain constant for all time in dimensions 2 and higher. In the case of a moving catalyst (gamma>0), we consider the solution u from the perspective of the catalyst, i.e., the expression u(t,Y_t+x). Focusing on the cases where moments grow exponentially fast (that is, gamma sufficiently large), we describe the moment asymptotics of the expression above up to equivalence. Here, it is crucial to prove the existence of a principal eigenfunction of the corresponding Hamilton operator. While this is well-established for the first moment, we have found an extension to higher moments.
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    A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Biskup, Marek; Salvi, Michele; Wolff, Tilman
    We consider resistor networks on Zd where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and arbitrary ellipticity contrasts are to be addressed in a subsequent paper.
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    Large deviations for the local times of a random walk among random conductances in a growing box
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) König, Wolfgang; Wolff, Tilman
    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 Zd. 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.