  We initiate the study of persistent homology of random geometric simplicial
complexes. Our main interest is in maximally persistent cycles of degree-$k$ in
persistent homology, for a either the \cech or the Vietoris--Rips filtration
built on a uniform Poisson process of intensity $n$ in the unit cube $[0,1]^d$.
This is a natural way of measuring the largest "$k$-dimensional hole" in a
random point set. This problem is in the intersection of geometric probability
and algebraic topology, and is naturally motivated by a probabilistic view of
topological inference.
  We show that for all $d \ge 2$ and $1 \le k \le d-1$ the maximally persistent
cycle has (multiplicative) persistence of order $$ \Theta
\left(\left(\frac{\log n}{\log \log n} \right)^{1/k} \right),$$ with high
probability, characterizing its rate of growth as $n \to \infty$. The implied
constants depend on $k$, $d$, and on whether we consider the Vietoris--Rips or
\cech filtration.
