Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models
| Autor: | Alessandra Faggionato, Hlafo Alfie Mimun |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Random graph Renormalization Boolean model Applied Mathematics 010102 general mathematics Poisson point process Miller–Abrahams random resistor network Left–right crossings Poisson distribution 01 natural sciences Combinatorics Symmetric function 010104 statistics & probability symbols.namesake Modeling and Simulation Bounded function symbols 0101 mathematics Connection (algebraic framework) Random variable Mathematics |
| Zdroj: | Stochastic Processes and their Applications. 137:62-105 |
| ISSN: | 0304-4149 |
| DOI: | 10.1016/j.spa.2021.03.001 |
| Popis: | 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 ≠ y and | x − y | ≤ 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 C n 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 ) γ with γ > 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect