scholarly journals Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

2020 ◽  
Author(s):  
Jonathan Hermon ◽  
Tom Hutchcroft
Keyword(s):  
Cluster Size ◽  
List Type ◽  
Point Function ◽  
Transitive Graph ◽  
Locally Finite ◽  
Infinite Cluster ◽  
Cambridge University ◽  
Percolation Probability ◽  
Discrete Potential Theory ◽  
Supercritical Phase

Abstract Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and $$p > p_c(G)$$ p > p c ( G ) then there exists a positive constant $$c_p$$ c p such that $$\begin{aligned} \mathbf {P}_p(n \le |K| < \infty ) \le e^{-c_p n} \end{aligned}$$ P p ( n ≤ | K | < ∞ ) ≤ e - c p n for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We deduce the following two corollaries: Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999). For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

scholarly journals Approximating graphs with polynomial growth

2000 ◽  
Vol 42 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Norbert Seifter ◽  
Wolfgang Woess
Keyword(s):  
Inverse Limit ◽  
Polynomial Growth ◽  
Infinite Sequence ◽  
Growth Conditions ◽  
Infinite Graph ◽  
Subject Classification ◽  
Transitive Graph ◽  
Locally Finite ◽  
Finite Graphs ◽  
Growth Properties

Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

scholarly journals Bounding basic characteristics of spatial epidemics with a new percolation model

2011 ◽  
Vol 43 (2) ◽  
pp. 335-347 ◽  
Author(s):  
Ronald Meester ◽  
Pieter Trapman
Keyword(s):  
Finite Graph ◽  
Weight Vector ◽  
Percolation Model ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Percolation Probability ◽  
Special Cases ◽  
Random Weight ◽  
Dependent Percolation ◽  
Basic Characteristics

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

scholarly journals Bounding basic characteristics of spatial epidemics with a new percolation model

2011 ◽  
Vol 43 (02) ◽  
pp. 335-347 ◽  
Author(s):  
Ronald Meester ◽  
Pieter Trapman
Keyword(s):  
Finite Graph ◽  
Weight Vector ◽  
Percolation Model ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Percolation Probability ◽  
Special Cases ◽  
Random Weight ◽  
Dependent Percolation ◽  
Basic Characteristics

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.

Thomas Brampton's Metrical Paraphrase of the Seven Penitential Psalms: A Diplomatic Edition of the Version in MS Pepys 1584 and MS Cambridge University Ff 2.38 with Variant Readings from All Known Manuscripts

Traditio ◽  
1951 ◽  
Vol 7 ◽  
pp. 359-403
Author(s):  
James R. Kreuzer
Keyword(s):  
Trinity College ◽  
List Type ◽  
Type Number ◽  
Cambridge University ◽  
Penitential Psalms ◽  
Diplomatic Edition

Thomas Brampton's Metrical Paraphrase of the Seven Penitential Psalms appears in the following manuscripts: SSloane 1853, fol. 3a. Century XVPPepys 2030, fol. 1a. (Begins at v 7 of stanza 7; lacks stanzas 121-124) Century XVIHHarley 1704, fol. 13a. (Stanzas 62-116 only) Century XVCCambridge University, Ff 2.38, fol. 28a. (Begins imperfectly in stanza 56). Century XVCvTrinity College, Cambridge 600, pp. 197-232. Century XVPvPepys 1584, Art. 3. Century XV

scholarly journals Planar Transitive Graphs

10.37236/7888 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Matthias Hamann
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Transitive Graph ◽  
Locally Finite ◽  
Transitive Graphs ◽  
Finitely Presented

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible. 

More About Robert of Flamborough's Penitential

Traditio ◽  
1961 ◽  
Vol 17 ◽  
pp. 531-532 ◽  
Author(s):  
Francis Firth
Keyword(s):  
List Type ◽  
Type Number ◽  
Cambridge University ◽  
Corpus Christi

I wish to add some details to my paper published in this Bulletin last year and also make a few corrections.Seven additional MSS have been examined.(1) MSS of Form A: Cc Cambridge, Corpus Christi College Libr. 441, pages 37-134;Ci Cambridge, University Libr. Ii.6.18, fols. 3r-129v;Ck Cambridge, University Libr. Kk.6.1, the whole MS, fols. 1r-97v;H Paris, Bibl, Nat. lat. 10691, the whole MS, fols. 1r-96r;Tu Tübingen, Deposit from the Prussian State Libr. of Berlin lat. fol, 212, fols. 56r-65r;U Paris, University Libr. 1247, fols. 10r-33r.

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