  Let integer $n \ge 3$ and integer $r = r(n) \ge 3$. Define the binomial
random $r$-uniform hypergraph $H_r(n, p)$ to be the $r$-uniform graph on the
vertex set $[n]$ such that each $r$-set is an edge independently with
probability $p$. A hypergraph is linear if every pair of hyperedges intersects
in at most one vertex. We study the probability of linearity of random
hypergraphs $H_r(n, p)$ via cluster expansion and give more precise asymptotics
of the probability in question, improving the asymptotic probability of
linearity obtained by McKay and Tian, in particular, when $r=3$ and $p =
o(n^{-7/5})$.
