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Monotonicity of conditional distributions and growth models on trees. (English) Zbl 1044.60094
Summary: We consider a sequence of probability measures \(\nu_n\) obtained by conditioning a random vector \(X=(X_1,\dots,X_d)\) with nonnegative integer valued components on \(X_1+\cdots+X_d=n-1\) 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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