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October 2000 Monotonicity of conditional distributions and growth models on trees
Thomas M. Liggett
Ann. Probab. 28(4): 1645-1665 (October 2000). DOI: 10.1214/aop/1019160501


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.


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Thomas M. Liggett. "Monotonicity of conditional distributions and growth models on trees." Ann. Probab. 28 (4) 1645 - 1665, October 2000.


Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60094
MathSciNet: MR1813837
Digital Object Identifier: 10.1214/aop/1019160501

Primary: 60K35

Keywords: Catalan numbers , contact process , coupling , Critical exponents , critical values , growth models on trees , Stochastic monotonicity

Rights: Copyright © 2000 Institute of Mathematical Statistics

Vol.28 • No. 4 • October 2000
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