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31
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 ..."
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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.
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.
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 ..."
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Cited by 12 (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.
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 11 (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
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points
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The survival probability for critical spreadout oriented percolation above 4 + 1 dimensions. II. Expansion
, 2005
"... We consider critical spreadout oriented percolation above 4+1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n+ 1) decays like 1/Bn2 as n→∞, where B is a fin ..."
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Cited by 7 (4 self)
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We consider critical spreadout oriented percolation above 4+1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n+ 1) decays like 1/Bn2 as n→∞, where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn as n→∞. The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with pointtoplane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction. 1 Introduction and
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.