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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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##### References:
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