Size dependence of the largest distance between random points

Abstract

A set of $N$ points is chosen randomly in a $D$-dimensional volume $V=a^D$, with periodic boundary conditions. For each point $i$, its distance $d_i$ is found to its nearest neighbour. Then, the maximal value is found, $d_{max}=max(d_i,i=1,...,N)$. Our numerical calculations indicate, that when the density $N/V$=const, $d_{max}$ scales with the linear system size as $d_{max}^2\propto a^{\varphi }$, with $\varphi =0.24\pm 0.04$ for $D=1,2,3,4$.

4 pages, 1 figure