# Student Probability Seminar

#### Nearest Neighbour Percolation and Extensions

Speaker: Elias Hess-Childs, CIMS

Location: Warren Weaver Hall 1314

Date: Monday, February 24, 2020, 11:45 a.m.

Synopsis:

Connect every point of a homogeneous poisson point process in R^d to its k-nearest neighbours to form a random graph G(d,k). With what probability does G(d,k) have an infinite connected component (i.e. when does G(d,k) percolate)?

The answer: when k=2, as long as d is sufficiently large.

Indeed, in their original paper, Meester and Häggström showed this interesting result by exploiting the geometry of large dimensional Euclidean space and using a local approximation.

In this talk I will introduce the nearest neighbour model, outline the techniques used in the original paper, then introduce a natural extension and a better context to discuss similar random objects.