Concentration inequalities and large deviations for continuous greedy animals and paths

Abstract

Consider the continuous greedy paths model: given a $d$-dimensional Poisson point process with positive marks interpreted as masses, let $\mathrm P(\ell)$ denote the maximum mass gathered by a path of length $\ell$ starting from the origin. It is known that $\mathrm P(\ell)/\ell converges a.s.\ to a deterministic constant $\mathrm P$. We show that the lower-tail deviation probability for $\mathrm P(\ell) has order $\mathrm{exp}(-\ell^2)$ and, under exponential moment assumption on the mass distribution, that the upper-tail deviation probability has order $\mathrm{exp}(-\ell)$. In the latter regime, we prove the existence and some properties -notably, convexity -of the corresponding rate function. An immediate corollary is the large deviation principle at speed $\ell$ for $\mathrm P(\ell)$. Along the proof we show an upper-tail concentration inequality in the case where marks are bounded. All of the above also holds for greedy animals and have versions where the paths or animals involved have two anchors instead of one.