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The Art of Random Walks
 Lecture Notes in Mathematics 1885
, 2006
"... 1.1 Basic definitions and preliminaries................ 8 ..."
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Cited by 15 (5 self)
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1.1 Basic definitions and preliminaries................ 8
The AlexanderOrbach conjecture holds in high dimensions
 Invent. Math
"... Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where meanfield behavior have been established, namely when the dimension d is large enough or when d> 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous di ..."
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Cited by 10 (2 self)
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Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where meanfield behavior have been established, namely when the dimension d is large enough or when d> 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension ds = 4 3, that is, pt(x,x) = t−2/3+o(1). This establishes a conjecture of Alexander and Orbach [4]. En route we calculate the onearm exponent with respect to the intrinsic distance. 1.
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 percol ..."
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Cited by 3 (1 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.
A C ∗ algebra of geometric operators on selfsimilar CWcomplexes. Novikov–Shubin and L 2 Betti numbers, preprint
, 2006
"... Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for su ..."
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Cited by 2 (2 self)
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Abstract. A class of CWcomplexes, called selfsimilar complexes, is introduced, together with C ∗algebras Aj of operators, endowed with a finite trace, acting on squaresummable cellular jchains. Since the Laplacian ∆j belongs to Aj, L 2Betti numbers and NovikovShubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the EulerPoincaré characteristic is proved. L 2Betti and NovikovShubin numbers are computed for some selfsimilar complexes arising from selfsimilar fractals. 1. Introduction. In this paper we address the question of the possibility of extending the definition of some L 2invariants, like the L 2Betti numbers and NovikovShubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe [29], who defined a trace
The Einstein relation for random walks on graphs
, 2008
"... This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the ..."
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This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic vwork for the study of (sub) diffusive behavior of the random walks on weighted graphs. 1
Upper Bounds for Transition Probabilities on Graphs
, 2004
"... In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities. ..."
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In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities.
Lower Bounds for Transition Probabilities on Graphs
, 2003
"... In this paper twosided estimate of the distribution of the exit time is shown together with off diagonal heat kernel lower bound for random walks on weighted graphs. ..."
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In this paper twosided estimate of the distribution of the exit time is shown together with off diagonal heat kernel lower bound for random walks on weighted graphs.
Markov Processes on Vermiculated Spaces
"... A general technique is given for constructing new Markov processes from existing ones. The new process and its state space are both projective limits of sequences built by an iterative scheme. The space at each stage in the scheme is obtained by taking disjoint copies of the space at the previous st ..."
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A general technique is given for constructing new Markov processes from existing ones. The new process and its state space are both projective limits of sequences built by an iterative scheme. The space at each stage in the scheme is obtained by taking disjoint copies of the space at the previous stage and quotienting to identify certain distinguished points. Away from the distinguished points, the process at each stage evolves like the one constructed at the previous stage on some copy of the previous state space, but when the process hits a distinguished point it enters at random another of the copies "pinned" at that point. Special cases of this construction produce diffusions on fractallike objects that have been studied recently.