Monotonicity of conditional distributions and growth models on trees

We consider a sequence of probability measures v(n) obtained by conditionin g a random vector X = (X-1,... ,X-d) with nonnegative integer valued components on X-1 +(...)+ X-d = n - 1 and give several sufficient conditions on the distribution of X for v(n) to be stochastically increasing in n. The problem is motivated by an interact ing particle system on the homogeneous tree in which each vertex has d + 1 neighbors. This system is a variant of the contact process and was studied recently by A. Puha. She showed that the critical value for this process is 1/4 if d = 2 and gave a conjectured expression for the critical value for all d. Our results confirm her conjecture, by showing that certain v(n)'s d efined in terms of d-ary Catalan numbers are stochastically increasing in n . The proof uses certain combinatorial identities satisfied by the d-ary Ca talan numbers.