The Lightning Model
Autor: | James T. Campbell, Alexandra Deane, Anthony Quas |
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Rok vydání: | 2021 |
Předmět: |
Statistics and Probability
Vertex (graph theory) Strongly connected component 010504 meteorology & atmospheric sciences General Mathematics 010102 general mathematics Probability (math.PR) Integer lattice Directed graph 01 natural sciences Standard Model Combinatorics 60K35 Percolation FOS: Mathematics Interval (graph theory) Almost surely 0101 mathematics Statistics Probability and Uncertainty Mathematics - Probability 0105 earth and related environmental sciences Mathematics |
DOI: | 10.48550/arxiv.2101.09383 |
Popis: | We introduce a non-standard model for percolation on the integer lattice $\mathbb Z^2$. Randomly assign to each vertex $a \in \mathbb Z^2$ a potential, denoted $\phi_a$, chosen independently and uniformly from the interval $[0, 1]$. For fixed $\epsilon \in [0,1]$, draw a directed edge from vertex $a$ to a nearest-neighbor vertex $b$ if $\phi_b < \phi_a + \epsilon$, yielding a directed subgraph of the infinite directed graph $\overrightarrow{G}$ whose vertex set is $\mathbb Z^2$, with nearest-neighbor edge set. We define notions of weak and strong percolation for our model, and observe that when $\epsilon = 0$ the model fails to percolate weakly, while for $\epsilon = 1$ it percolates strongly. We show that there is a positive $\epsilon_0$ so that for $0 \le \epsilon \le \epsilon_0$, the model fails to percolate weakly, and that when $\epsilon > p_\text{site}$, the critical probability for standard site percolation in $\mathbb Z^2$, the model percolates strongly. We study the number of infinite strongly connected clusters occurring in a typical configuration. We show that for these models of percolation on directed graphs, there are some subtle issues that do not arise for undirected percolation. Although our model does not have the finite energy property, we are able to show that, as in the standard model, the number of infinite strongly connected clusters is almost surely 0, 1 or $\infty$. Comment: To appear: J. Theor. Prob |
Databáze: | OpenAIRE |
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