Extreme Value Distribution for the Largest Cube in a Random Lattice

Abstract

Suppose that the sites of a finite d-dimensional lattice $(d\geqq 2)$ of side n are occupied by independent, identically distributed random variables with value 0 or 1. The length of the side of the largest cube of l’s is found to have (approximately) an integerized extreme Value distribution. The distribution becomes increasingly concentrated on three consecutive integers, as n increases. Applications to clustering are discussed.