Percolation

Grimmett, Geoffrey

doi:10.1007/978-3-0348-8968-1_18

Cited by 97 publications

(169 citation statements)

References 52 publications

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“…For percolation on graphs it is known that there is a critical probability 0 < p c < 1 such that for p < p c τ (x, y) decays exponentially in the distance |x − y| and for p > p c it does not. This is also related to the occurrence of an infinite connected cluster above p c with probability one, Grimmett (1999).…”

Section: Some Consequences From Finite Volume Criteriamentioning

confidence: 99%

A Short Introduction to Anderson Localization

Hundertmark¹

2008

Analysis and Stochastics of Growth Processes and Interface Models

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Section: Some Consequences From Finite Volume Criteriamentioning

confidence: 99%

A Short Introduction to Anderson Localization

Hundertmark¹

2008

Analysis and Stochastics of Growth Processes and Interface Models

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“…So let us start with the relevant percolation-type estimates; for background on percolation theory see [15]. Consider site percolation on Z d with P(v open) = q independently for every v ∈ Z d .…”

Section: Dimension Higher Than Onementioning

confidence: 99%

Localisation in the Bouchaud–Anderson model

Muirhead

Pymar

2016

Stochastic Processes and their Applications

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Abstract. It is well-known that both random branching and trapping mechanisms can induce localisation phenomena in random walks; the prototypical examples being the parabolic Anderson and Bouchaud trap models respectively. Our aim is to investigate how these localisation phenomena interact in a hybrid model combining the dynamics of the parabolic Anderson and Bouchaud trap models. Under certain natural assumptions, we show that the localisation effects due to random branching and trapping mechanisms tend to (i) mutually reinforce, and (ii) induce a local correlation in the random fields (the 'fit and stable' hypothesis of population dynamics). Minor revision to published version. This is an updated version of [23] containing the following minor revisions:• A typo in the statement of Proposition 3.14 has been corrected;• The coupling used in Section 5 has been slightly modified to fix a gap in its original statement; we thank Renato Soares dos Santos for pointing this out to us; • A slight correction has been made to the proof of Proposition 4.11; and • The bibliography has been updated.

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“…For convenience, we assume that the greatest common divisor gcd{k : p k > 0} = 1. The condition ensures that the infinite open cluster is unique (see [5,Theorem 12.3]). If gcd{k : p k > 0} = m > 1 then we consider the bond percolation on mZ 2 with…”

Section: Proofsmentioning

confidence: 99%