The Coincidence of Critical Points in Poisson Percolation Models

Menshikov, Mikhail; Sidorenko, Alexander

doi:10.1137/1132083

Cited by 140 publications

(211 citation statements)

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“…So, since the susceptibility is monotone increasing, it follows that (1) holds for all t t b . Combining our findings, Lemma 4,(16) and (17) now yield Theorem 1 with ρ k (t) = ϕ(k, t).…”

Section: Proof Of Theoremsupporting

confidence: 71%

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The evolution of subcritical Achlioptas processes

Riordan

Warnke

2014

Random Struct Algorithms

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In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. Although the evolution of such 'local' modifications of the Erdős-Rényi random graph process has received considerable attention during the last decade, so far only rather simple rules are well understood. Indeed, the main focus has been on 'bounded-size' rules, where all component sizes larger than some constant B are treated the same way, and for more complex rules very few rigorous results are known.In this paper we study Achlioptas processes given by (unbounded) size rules such as the sum and product rules. Using a variant of the neighbourhood exploration process and branching process arguments we show that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges. Our convergence result is most likely best possible for certain rules: in the later evolution the number of vertices in small components may not be concentrated. Furthermore, we believe that for a large class of rules the critical time where the susceptibility 'blows up' coincides with the percolation threshold.

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“…So, since the susceptibility is monotone increasing, it follows that (1) holds for all t t b . Combining our findings, Lemma 4,(16) and (17) now yield Theorem 1 with ρ k (t) = ϕ(k, t).…”

Section: Proof Of Theoremsupporting

confidence: 71%

“…We first assume that G R tj−1n satisfies the induction hypothesis, i.e., (19)- (22) with j replaced by j − 1. In particular, (16) and (17) hold for t = t j−1 with r t = j − 1, and we have S(…”

Section: Proof Of Theoremmentioning

confidence: 99%

The evolution of subcritical Achlioptas processes

Riordan

Warnke

2014

Random Struct Algorithms

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“…It was only in 1980, twenty years after Harris' proof of the inequality p H ≥ 1/2, that Kesten [16] proved that p T = p H = 1/2. Since then, Menshikov [20] (see also Menshikov, Molchanov and Sidorenko [21]) and Aizenman and Barsky [1] (see also Grimmett [10]) have shown that p T = p H in great generality, in particular, for site percolation in any lattice graph; see Section 4.1 for a formal definition. Note that bond percolation in a lattice graph L corresponds to site percolation in the line graph of L, which can be realized as a lattice graph, so results for site percolation in general lattices apply to bond percolation as well.…”

Section: Introductionmentioning

confidence: 99%

Sharp thresholds and percolation in the plane

Bollobás

Riordan

2006

Random Struct Algorithms

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Recently, it was shown in [4] that the critical probability for random Voronoi percolation in the plane is 1/2. As a by-product of the method, a short proof of the Harris-Kesten Theorem was given in [5]. The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results, and to prove new ones.

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“…We prove Theorem 6.10 via Menshikov's method 268,271] rather than that of Aizenman{Barsky 13]. The proof given below is essentially a reproduction of that given in G], but with the correction of a minor error on page 50 of G].…”

Section: Site and Bond Percolationmentioning

confidence: 99%

“…Subsequently, it was proved in 13, 268,271] that this hypothesis is satis ed whenever p < p c . The situation was similar for supercritical percolation, the corresponding hypothesis being that percolation occurs in slabs.…”

Section: Percolation In Slabsmentioning

confidence: 99%

Percolation and disordered systems

Grimmett

1997

Lecture Notes in Mathematics

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PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematical theory of percolation and related topics. The rst nine chapters summarise rigorous results in percolation theory, with special emphasis on results obtained since the publication of my book 156] entitled`Percolation', and sometimes referred to simply as G] in these notes. Following this core material are chapters on random walks in random labyrinths, and fractal percolation. The nal two chapters include material on Ising, Potts, and random-cluster models, and concentrate on a`percolative' approach to the associated phase transitions.The rst target of this course is to draw a picture of the mathematics of percolation, together with its immediate mathematical relations. Another target is to present and summarise recent progress. There is a considerable overlap between the rst nine chapters and the contents of the principal reference G]. On the other hand, the current notes are more concise than G], and include some important extensions, such as material concerning strict inequalities for critical probabilities, the uniqueness of the in nite cluster, the triangle condition and lace expansion in high dimensions, together with material concerning percolation in slabs, and conformal invariance in two dimensions. The present account di ers from that of G] in numerous minor ways also. It does not claim to be comprehensive. A second edition of G] is planned, containing further material based in part on the current notes.A special feature is the bibliography, which is a fairly full list of papers published in recent years on percolation and related mathematical phenomena. The compilation of the list was greatly facilitated by the kind responses of many individuals to my request for lists of publications.Many people have commented on versions of these notes, the bulk of which have been typed so superbly by Sarah Shea-Simonds. I thank all those who have contributed, and acknowledge particularly the suggestions of Ken Alexander, Carol Bezuidenhout, Philipp Hiemer, Anthony Quas, and Alan Stacey, some of whom are mentioned at appropriate points in the text. In addition, these notes have bene ted from the critical observations of various members of the audience at St Flour.Members of the 1996 summer school were treated to a guided tour of the library of the former seminary of St Flour. We were pleased to nd there a copy of the Encyclop edie, ou Dictionnaire Raisonn e des Sciences, des Arts et des M etiers, compiled by Diderot and D'Alembert, and published in Geneva around 1778. Of the many illuminating entries in this substantial work, the following de nition of a probabilist was not overlooked. 3 PROBABILISTE, s. m. (Gram. Th eol.) celui qui tient pour la doctrine abominable des opinions rendues probables par la d ecision d'un casuiste, & qui assure l'innocence de l'action faite en cons equence. Pascal a foudroy e ce systême, qui ouvroit la porte au crime en accordant a l'autorit e les pr erogatives de la certitude, a l'opinion & la s ec...

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The Coincidence of Critical Points in Poisson Percolation Models

Cited by 140 publications

References 5 publications

The evolution of subcritical Achlioptas processes

The evolution of subcritical Achlioptas processes

Sharp thresholds and percolation in the plane

Percolation and disordered systems

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