Left-right crossings in the Miller-Abrahams random resistor network and in generalized Boolean models

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:137

Article

Faggionato, Alessandra¹  Mimun, Hlafo Alfie² 

[1] Univ Roma La Sapienza, Dept Math, Ple Aldo Moro 2, I-00185 Rome, Italy

[2] LUISS Guido Carli, Dept Econ & Finance, Viale Romania 32, I-00197 Rome, Italy

关键词: Poisson point process;    Boolean model;    Miller-Abrahams random resistor network;    Left-right crossings;    Renormalization;   

DOI  :  10.1016/j.spa.2021.03.001

来源: Elsevier

【 摘 要 】

We consider random graphs G built on a homogeneous Poisson point process on R-d, d >= 2, with points x marked by i.i.d. random variables E-x. Fixed a symmetric function h(., .), the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs {x, y} with x not equal y and vertical bar x - y vertical bar <= h(E-x, E-y). We call G Poisson h-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h(a, b) := a + b and E-x >= 0. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left-right crossings in a box of size n is lower bounded by Cn(d-1) apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h(a, b) = (a + b)(gamma) with gamma > 0, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller-Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

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