The phase transitions of diameters in random axis-parallel hyperrectangle intersection graphs

Abstract

We study the behaviours of diameters in two models of random axis-parallel hyperrectangle intersection graphs: $G(n,r,l)$ and $G_u(n,r,l)$. These two models use axis-parallel $l$-dimensional hyperrectangles to represent vertices, and vertices are adjacent if and only if their corresponding axis-parallel hyperrectangles intersect. In the model $G(n,r,l)$, we distribute $n$ points within ${[0,1]}^l$ uniformly and independently, and each point is the centre of an axis-parallel $l$-dimensional hypercube with edge length $r$. The model $G_u(n,r,l)$, distributing the centres of $n$ axis-parallel $l$-dimensional hyperrectangles within ${[0,1]}^l$ exactly as the model $G(n,r,l)$, assigns a length from a uniform distribution over $[0,r]$ to each edge of the $n$ axis-parallel $l$-dimensional hyperrectangles.

We prove that in the model $G(n,r,l)$, there is a phase transition for the event that the diameter is at most $d\left(n\right)$ occurring at $r=d{\left(n\right)}^{-1}$ if $n\cdot d{\left(n\right)}^{-l\left(n\right)}\ge n^{ϵ},$ where $0<ϵ<1$ is an arbitrary small constant, and $l=l\left(n\right)=o\left(1\right)\cdot (lnn)\cdot {(lnlnn)}^{-1}.$ In the model $G_u(n,r,l)$, this phase transition occurs at $r={\left(d\left(n\right)-1\right)}^{-1}$.