Upper large deviations for power-weighted edge lengths in spatial random networks

Abstract

We study the large-volume asymptotics of the sum of power-weighted edge lengths ${\sum }_{e\in E}|e{|}^{\alpha }$ in Poisson-based spatial random networks. In the regime $\alpha >d$, we provide a set of sufficient conditions under which the upper large deviations asymptotics are characterized by a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. Moreover, the rate function can be expressed through a concrete optimization problem. This framework encompasses in particular directed, bidirected and undirected variants of the $k$-nearest neighbor graph, as well as suitable $\beta$-skeletons.

30 pages, 5 figures

References in corpus (2)

• Semi-infinite paths of the 2d-Radial Spanning Tree
• A variational formula for the free energy of an interacting many-particle system

Cited by in corpus (1)

• Lower large deviations for geometric functionals in sparse, critical and dense regimes