The INI has a new website!

This is a legacy webpage. Please visit the new site to ensure you are seeing up to date information.

Skip to content

SCS

Seminar

Strict inequalities of critical points in continuum percolation

Penrose, M (Bath)
Friday 09 April 2010, 09:30-10:30

Seminar Room 1, Newton Institute

Abstract

For any infinite connected graph, the critical probabilities for bond percolation and for site percolation satisfy the inequality $p_c^{\rm bond} \leq p_c^{\rm site}$. Moreover, this is known to be a strict inequality on a large class of lattices. In this talk, we discuss the extension of the strict inequality to certain {\em random} graphs arising in continuum percolation, including the Gilbert graph in which each point of a homogeneous planar Poisson point process of supercritical intensity $\lambda$ is connected by an edge to every other Poisson point within unit distance. More generally, we consider the random connection model, in which each pair of Poisson points distant at most $r$ apart is connected by an edge with probability $p$, so that the average node degree is $\lambda \pi r^2 p$. Given $r$ and $p$ there is a critical intensity $\lambda_c(p,r)$ above which the graph percolates. As well as the strict inequality $p_c^{\rm bond} < p_c^{\rm site}$ for this graph, we discuss a related result which says that $\lambda_c(r^{-2},r)$ is strictly decreasing in $r $. That is, for a given average node degree it is easier to percolate in a graph with long-range connections than in a graph with only short-range connections.

Presentation

[pdf ]

Video

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.

Back to top ∧