Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs

Antunović, Tonći; Veselić, Ivan

doi:10.1007/s10955-007-9459-x

Cited by 31 publications

(48 citation statements)

References 12 publications

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“…. , |Q| are given by (4). Thus the set 1 (δ, R, Q) consists of all configurations where the number of edges of length longer than R that are incident to at least one vertex in Q is at least |Q|(ε(R) + δ).…”

Section: Lemma 32mentioning

confidence: 99%

Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

Schwarzenberger

2012

J Stat Phys

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In this paper we study the spectrum of long-range percolation graphs. The underlying geometry is given in terms of a finitely generated amenable group. We prove that the integrated density of states (IDS) or spectral distribution function can be approximated uniformly in the energy variable. This result is already new for percolation on Z d . Using this, we are able to characterize the set of discontinuities of the IDS.

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Section: Lemma 32mentioning

confidence: 99%

Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

Schwarzenberger

2012

J Stat Phys

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“…Invariant percolation on the Euclidean lattice as well as on homogeneous trees in the subcritical regime shows to have a cluster size |C o | with exponentially decaying distribution [1] (see the generalisations of this to quasi-transitive graphs in [3]). …”

Section: Application To the Expected Return Probabilitymentioning

confidence: 91%

An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters

Sobieczky

2010

J Theor Probab

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A comparison technique for random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition probabilities is found. In this case, the state space is a random partial graph of a regular graph of bounded geometry and transitive automorphism group. The law of the random edge-set is assumed to be stationary with respect to some transitive subgroup of the automorphism group ('invariant percolation'). Given that this subgroup is unimodular, it is shown that stationarity strengthens the upper bound of the expected return probability, compared with standard bounds derived from the Cheeger inequality.

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“…, p N −1 ) and q = p N ), (2.1) (Note that ψ p,q (x, S) is the analogue of φ p (S) defined in [4] for homogeneous percolation on transitive graphs.) We define the ψ-critical surface 2) or equivalently,…”

Section: Outline Of the Proofmentioning

confidence: 99%

Sharpness for inhomogeneous percolation on quasi-transitive graphs

Beekenkamp

Hulshof

2019

Statistics & Probability Letters

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In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive graphs by , and the result generalizes previous results of Antunović and Veselić [2] and Menshikov [7].

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Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs

Cited by 31 publications

References 12 publications

Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters

Sharpness for inhomogeneous percolation on quasi-transitive graphs

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