Results 1  10
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24
The spectral dimension of generic trees
"... Abstract. We define generic ensembles of infinite trees. These are limits as N → ∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability m ..."
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Cited by 14 (8 self)
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Abstract. We define generic ensembles of infinite trees. These are limits as N → ∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension dh = 2. Our main result is that their spectral dimension is ds = 4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the twopoint function along the spine, is
Critical random graphs: diameter and mixing time
"... Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Worm ..."
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Cited by 13 (4 self)
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Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of pbond percolation on any dregular nvertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.
Anomalous heatkernel decay for random walk among bounded random conductances
, 2008
"... ABSTRACT. We consider the nearestneighbor simple random walk on Z d, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy> 0 exceeds the threshold for bond percolation on Z d. For environments in whi ..."
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Cited by 13 (2 self)
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ABSTRACT. We consider the nearestneighbor simple random walk on Z d, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy> 0 exceeds the threshold for bond percolation on Z d. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2nstep return probability P 2n ω (0,0). We prove that P 2n ω (0,0) is bounded by a random constant times n −d/2 in d = 2,3, while it is o(n −2) in d ≥ 5 and O(n −2 log n) in d = 4. By producing examples with anomalous heatkernel decay approaching 1/n 2 we prove that the o(n −2) bound in d ≥ 5 is the best possible. We also construct natural ndependent environments that exhibit the extra log n factor in d = 4. 1.
The mixing time of the giant component of a random graph
"... We show that the total variation mixing time of the simple random walk on the giant component of supercritical G(n,p) and G(n,m) is Θ(log 2 n). This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is intere ..."
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Cited by 10 (4 self)
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We show that the total variation mixing time of the simple random walk on the giant component of supercritical G(n,p) and G(n,m) is Θ(log 2 n). This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are “decorated expanders ” — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. 1
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.
Heat kernel estimates for strongly recurrent random walk on random media, preprint. Available from http://www.math.kyotou.ac.jp/ ∼kumagai/KumaMisumi.pdf
, 2007
"... random media ..."
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Invasion percolation on regular trees
, 2006
"... We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its rpoint function for any r ≥ 2 and of its volume both at a given height and below a given height. In addition, we derive scaling estimates for simple ra ..."
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Cited by 5 (1 self)
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We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its rpoint function for any r ≥ 2 and of its volume both at a given height and below a given height. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay towards the critical value for ordinary percolation, but only slowly, and this slow decay causes an anomalous scaling behaviour. MSC 2000. 60K35, 82B43 Key words and phrases. Invasion percolation cluster, incipient infinite cluster, rpoint function, cluster size, simple random walk, Poisson point process.
Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree
, 2008
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A note on percolation on Z d : Isoperimetric profile via exponential cluster repulsion
, 2007
"... Abstract. We show that for all p> pc(Z d) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the iso ..."
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Cited by 2 (0 self)
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Abstract. We show that for all p> pc(Z d) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. This implies for instance that simple random walk on the largest cluster of a finite box [−n,n] d with high probability has L ∞mixing time Θ(n 2), and that the heat kernel (return probability) on the infinite cluster a.s. decays like pn(o, o) = O(n −d/2). Versions of these results have been proven by Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau (2006). We also give a short proof of a theorem of Angel, Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedge in Z 3 is a.s. transient whenever the wedge itself is transient. 1. Introduction and