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57
Random walk on the incipient infinite cluster on trees
 Illinois J. Math
"... 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 ..."
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Cited by 63 (16 self)
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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
Characterization of subGaussian heat kernel . . .
, 2000
"... SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. ..."
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Cited by 43 (8 self)
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SubGaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities.
Which Values of the Volume Growth and Escape Time Exponent Are Possible for a Graph?
, 2001
"... Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, 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 ..."
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Cited by 43 (8 self)
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Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, 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.
Stability of parabolic Harnack inequalities
 Transactions of the American Mathematical Society
, 2004
"... Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. ..."
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Cited by 30 (4 self)
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Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. We also give stable necessary and sucient conditions for this parabolic Harnack inequality to hold.
DECOUPLING INEQUALITIES AND INTERLACEMENT PERCOLATION ON G × Z
, 2010
"... We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the per ..."
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Cited by 23 (4 self)
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We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percolation of the vacant set of random interlacements is always finite in our setup, and that it is positive when α ≥ 1 + β 2. We also obtain several stretched exponential controls both in the percolative and nonpercolative phases of the model. Even in the case where G = Zd, d ≥ 2, several of these results are new.
The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 20 (4 self)
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1.1 Basic definitions and preliminaries................ 8
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps
, 2008
"... We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the c ..."
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Cited by 19 (8 self)
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
Pointwise estimates for transition probabilities of random walks in infinite graphs
 in: Trends in mathematics: Fractals in Graz 2001
, 2002
"... walks on infinite graphs ..."
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