Universal aspects of critical percolation on random half-planar maps

Richier, Loïc

doi:10.1214/ejp.v20-4041

Cited by 20 publications

(26 citation statements)

References 25 publications

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“…Remark 1.1. Shortly after making this work public there appeared a paper by Loïc Richier [22] in which he (amongst many other things) determined the exact value of the percolation threshold to be p c = 5 9 , a stronger result than our bounds given in Proposition 1.4 and (1.10) below. His method does not involve analyzing an 'evoloving boundary condition' X n as ours do, and is technically simpler.…”

Section: Introductionmentioning

confidence: 74%

On Site Percolation in Random Quadrangulations of the Half-Plane

Björnberg

Stefánsson

2015

J Stat Phys

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We study site percolation on uniform quadrangulations of the upper half plane. The main contribution is a method for applying Angel's peeling process, in particular for analyzing an evolving boundary condition during the peeling. Our method lets us obtain rigorous and explicit upper and lower bounds on the percolation threshold pc, and thus show in particular that 0.5511 ≤ pc ≤ 0.5581. The method can be extended to site percolation on other half-planar maps with the domain Markov property.

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Section: Introductionmentioning

confidence: 74%

On Site Percolation in Random Quadrangulations of the Half-Plane

Björnberg

Stefánsson

2015

J Stat Phys

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“…for all λ > 0. In order to evaluate the expectation value on the left-hand side let us write g = exp(−λl −2 /B ν ) < 1 and consider the disk function W (l) g in (27). Recall that we may identify W (l) g (q) = g 1+l/2 W (l) (q g ) where q g is the weight sequence determined by (q g ) k := g (k−2)/2 q k and let c ± (g) (and r(g) = −c − (g)/c + (g)) be the associated constants from Proposition 2.…”

Section: Scaling Limit Of the Volume Processmentioning

confidence: 99%

“…Nevertheless, the peeling procedure in various forms received renewed interest in the mathematical literature, starting with its formalization in the setting of the uniform infinite planar triangulation (UIPT, [6]) by Angel in [4]. It was recognized that peeling could not only be used to study geometry, but that due to its important Markov properties it could be tailored towards studying many other aspects of random maps, including various forms of percolation [4,5,25,1,27,10], random walks [7], and some aspects of their conformal structure [18].…”

Section: Introductionmentioning

confidence: 99%

The Peeling Process of Infinite Boltzmann Planar Maps

Budd

2016

Electron. J. Combin.

124

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We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissibility criterion has a very simple interpretation. The finite random planar maps under consideration were recently proved to possess a well-defined local limit known as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien and Le Gall, we show that the peeling process on the IBPM can be obtained from the peeling process of finite random maps by conditioning the perimeter process to stay positive. The simplicity of the resulting description of the peeling process allows us to obtain the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight sequences.1 The derivation of the two-point function in [3] has always been regarded as heuristic, leading to an approximate probability distribution of the graph distance between random vertices that only becomes exact in the scaling limit. Only recently in [1] it was recognized that the expressions in [3] are already exact in the discrete setting, but they do not correspond to the graph distance of the triangulation but to a first-passage time on the dual cubic map, which agree up to rescaling in the scaling limit.

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“…Indeed, it is often possible to use the spatial Markov property of the underlying lattice to define an exploration along the percolation interface and get access to the percolation threshold. This approach was first developed in the pioneer work of Angel [1] for site-percolation on the Uniform Infinite Planar Triangulation (UIPT) and later extended to other models of percolation and maps [2,10,20,24]. As opposed to the "dynamical" approach of the peeling process, the work [11] uses a "fixed" combinatorial decomposition (inspired by [5]) and known enumeration results on triangulations to study the scaling limit of percolation cluster conditioned on having a large boundary.…”

Section: Introductionmentioning

confidence: 99%

A Boltzmann Approach to Percolation on Random Triangulations

Bernardi¹,

Curien²,

Miermont

2019

Can. J. Math.-J. Can. Math.

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We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bondpercolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length n decays exponentially with n except at a particular value pc of the percolation parameter p for which the decay is polynomial (of order n −10/3 ). Moreover, the probability that the origin cluster has size n decays exponentially if p < pc and polynomially if p ≥ pc.The critical percolation value is pc = 1/2 for site percolation, and pc = 2

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Universal aspects of critical percolation on random half-planar maps

Cited by 20 publications

References 25 publications

On Site Percolation in Random Quadrangulations of the Half-Plane

On Site Percolation in Random Quadrangulations of the Half-Plane

The Peeling Process of Infinite Boltzmann Planar Maps

A Boltzmann Approach to Percolation on Random Triangulations

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