A Model of Random Merging of Segments

摘要

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 denned as sup{q | lim(n→∞) h_n = 0}. Lower and upper bounds for q_c(m) are obtained.