Model of random merging of segments

We consider a growing set $U$ of segments with integer endpoints on a line. For every pair of adjacent segments, their union is added to $U$ with probability $q$. At the beginning, $U$ contains all segments of length from $1$ to $m$. Let $h_n$ be the probability that the segment $[a,a+n]$ will be created; the critical value $q_c(m)$ is defined as $\sup\{q\mid\lim_{n\to\infty}h_n=0\}$. Lower and upper bounds for $q_c(m)$ are obtained.