Monotonicity of conditional distributions and growth models on trees

Abstract

We consider a sequence of probability measures $\nu_n$ obtained by conditioning a random vector $X =(X_1,\ldots,X_d)$ with nonnegative integer valued components on

$$ X_1 + \dots + X_d = n - 1 $$

and give several sufficient conditions on the distribution of $X$ for $\nu_n$ to be stochastically increasing in $n$. The problem is motivated by an interacting 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 $\nu_n$’s defined in terms of $d$-ary Catalan numbers are stochastically increasing in $n$. The proof uses certain combinatorial identities satisfied by the $d$-ary Catalan numbers.