We show that a fi-parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to so called fi-Gaussian estimates for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order R . The latter condition can be replaced by a certain estimate of a resistance of annuli.

Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ff-regular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simple random walk on \Gamma starting at x from the ball centre x and radius r. We say \Gamma has escape time exponent fi ? 0 if there exists a constant c such that c T (x; r) cr for r 1. Well known estimates for random walks on graphs imply that ff 1 and 2 fi 1 + ff.

Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4

We explore some of the connections between the local picture left by the trace of simple random walk on a cylinder (Z/NZ) d × Z, d ≥ 2, running for times of order N 2d and the model of random interlacements recently introduced in [10]. In particular we show that for large N in the neighborhood of a point of the cylinder with vertical component of order N d the complement of the set of points visited by the walk up to times of order N 2d is close in distribution to the law of the vacant set of [10] with a level which is determined by an independent Brownian local time. The limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived.

We consider random walk on a discrete torus E of side-length N, in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uN d. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const log N. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites N d of E, and any point of E lies within distance N β of this component, with β an arbitrary positive number. 1

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In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concerning the relation of coupling with the EHI.

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The vacant set of random interlacements on Z d, d ≥ 3, has non-trivial percolative properties. It is known from [18], [16], that there is a non-degenerate critical value u ∗ such that the vacant set at level u percolates when u < u ∗ and does not percolate when u> u∗. We derive here an asymptotic upper bound on u∗, as d goes to infinity, which complements the lower bound from [21]. Our main result shows that u ∗ is equivalent to log d for large d, and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in [23].