=+14. Branching processes can be used to model the formation of polymers

[154]. Consider a large batch of identical subunits in solution. Each subunit has m > 1 reactive sites that can attach to similar reactive sites on other subunits. For the sake of simplicity, assume that a polymer starts from a fixed ancestral subunit and forms a tree structure with no cross linking of existing subunits. Also assume that each reactive site behaves independently and bonds to another site with probability p. Subunits attached to the ancestral subunit form the first generation of a branching process. Subunits attached to these subunits form the second generation and so forth. In this problem we investigate the possibility that polymers of infinite size form. In this case the solution turns into a gel. Show that the progeny distribution 9.9 Problems 239 for the first generation is binomial with m trials and success probability p and that the progeny distribution for subsequent generations is binomial with m−1 trials and success probability p. Show that the extinction probability t∞ satisfies t∞ = (1 − p + ps∞)

m s∞ = (1 − p + ps∞)

m−1, where s∞ is the extinction probability for a line of descent emanating from a first-generation subunit. Prove that polymers of infinite size can occur if and only if (m − 1)p > 1.