The van den Berg–Kesten–Reimer operator and inequality for infinite spaces

Arratia, Richard; Garibaldi, Skip; Hales, Alfred W.

doi:10.3150/16-bej883

Cited by 13 publications

(25 citation statements)

References 17 publications

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“…for every p ∈ [0, 1] and every two increasing events A and B. Reimer's inequality [58] states that the same inequality holds for arbitrary events A and B. Both inequalities are usually stated for events depending on at most finitely many edges, but this assumption is shown to be unnecessary in [6]. We shall use Reimer's inequality only in the special case that A and B can each be written as the intersection of an increasing event and a decreasing event, which has a simpler and earlier proof due to van den Berg and Fiebig [68].…”

Section: The Harris-fkg and Bk Inequalitiesmentioning

confidence: 99%

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Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

Hutchcroft

2020

J. Amer. Math. Soc.

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We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty phase in which there are infinite light clusters, which implies the existence of a non-empty phase in which there are infinitely many infinite clusters. That is, we show that p c < p h ≤ p u for any such graph. This answers a question of Häggström, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values.All our results apply, for example, to the product T k ×Z d of a k-regular tree with Z d for k ≥ 3 and d ≥ 1, for which these results were previously known only for large k. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and Z d edges are given different retention probabilities. These features had only previously been established for d = 1, k large. * Statslab, DPMMS, University of Cambridge

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Section: The Harris-fkg and Bk Inequalitiesmentioning

confidence: 99%

“…(6.11) (Again, we refer the reader to[6] for a justification of Reimer's inequality in infinite volume. )LetK be the set of vertices that are connected to v by a path consisting of open edges none of which have both endpoints in L −∞,− , and let K be the set of edges have at least one endpoint in K and do not have both endpoints in L −∞,− .…”

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confidence: 99%

Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

Hutchcroft

2020

J. Amer. Math. Soc.

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“…An equivalent definition of the box operation in (2) in terms of cylinders [A] K of subsets A of S, appearing as (2) in [1] is…”

Section: Introductionmentioning

confidence: 99%

“…A version of the BKR inequality for a finite product of arbitrary probability spaces, including finite spaces, and discrete or non-discrete infinite spaces, was considered in [7]. The case where A and B are subsets of finite or countable products of R, and more generally of Polish spaces, was considered in [1]. This later work raises important issues regarding the measurability of A B.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, if A B is measurable, then it follows from (6) that P (A B) ≤ P (A)P (B). Lemma 1 in [1] shows that if A and B are Borel subsets of [0, 1] n then A B is Lebesgue measurable. As stated in Section 8 of [1], the same proof holds on the product of Polish subspaces of R equipped with the completion of any Borel product probability.…”

Section: Introductionmentioning

confidence: 99%

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A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Goldstein¹,

Rinott

2018

Methodol Comput Appl Probab

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Given events A and B on a product space S = n i=1 S i , the set A B consists of all vectors x = (x 1 , . . . , x n ) ∈ S for which there exist disjoint coordinate subsets K and L of {1, . . . , n} such that given the coordinates x i , i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x j , j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequalitywas conjectured by van den Berg and Kesten [3] and proved by Reimer [13]. In [7] inequality (1) was extended to general product probability spaces, replacing A B by the set A 11 B consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality (1) results, due to the fact that A B ⊆ A 11 B. In particular, it may be the case that A B is empty, while A 11 B is not.We propose the further extension A st B depending on probability thresholds s and t, where A 11 B is the special case where both s and t take the value one. The outcomes x in A st B are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

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Nonexistence of Bigeodesics in Planar Exponential Last Passage Percolation

Basu

Hoffman

Sly

2021

Commun. Math. Phys.

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Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics, however the result has not been proved in any case. For the exactly solvable model of directed last passage percolation on Z 2 with i.i.d. exponential passage times, we study the corresponding question and show that almost surely the only bigeodesics are the trivial ones, i.e., the horizontal and vertical lines. The proof makes use of estimates for last passage time available from the integrable probability literature to study coalescence structure of finite geodesics, thereby making rigorous a heuristic argument due to Newman [2].

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The van den Berg–Kesten–Reimer operator and inequality for infinite spaces

Cited by 13 publications

References 17 publications

Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Nonexistence of Bigeodesics in Planar Exponential Last Passage Percolation

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