Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diffusions, hold with constants c_i depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for q_t(x,\cdot) holds only for t\ge S_x(\omega), where the constant S_x(\omega) depends on the percolation configuration \omega.Comment: Published at http://dx.doi.org/10.1214/009117904000000748 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exponential tail bounds for loop-erased random walk in two dimensions

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent loop-erased random walk both started at the origin do not intersect up to leaving the ball of radius $n$. This allows us to show that there exists $C$ such that for all $n$ and all $k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and all $\lambda\geq0$, $$\mathbf{P}\{M_n>\lambda\mathbf{E}[M_n]\}\leq2e^{-c\lambda}.$$ Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $\alpha0$ such that for all $n$ and $\lambda>0$, $$\mathbf{P}\{M_n<\lambda^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'\lambda ^{\alpha}}.$$Comment: Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Energy inequalities for cutoff functions and some applications

We consider a metric measure space with a local regular Dirichlet form. We establish necessary and sufficient conditions for upper heat kernel bounds with sub-diffusive space-time exponent to hold. This characterization is stable under rough isometries, that is it is preserved under bounded perturbations of the Dirichlet form. Further, we give a criterion for stochastic completeness in terms of a Sobolev inequality for cutoff functions. As an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space, where the well-established criterion in terms of volume growth fails

The random conductance model with Cauchy tails

We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^{\omega}_{n^2t}(0,y)$ in [Ann. Probab. (2009). To appear], Theorem 5.14, to a result which gives uniform convergence for $p^{\omega}_{n^2t}(x,y)$ for all $x,y$ in a ball.Comment: Published in at http://dx.doi.org/10.1214/09-AAP638 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Restoration of isotropy on fractals

We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian field theories. The mechanism is unique in that it is absent in spaces with translational invariance, while universal in that it is observed in a wide class of fractals.Comment: 11 pages, REVTEX, 3 postscript figures. (Compressed and encoded figures archived by "figure" command). To appear in Physical Review Letter