Monotonicity of conditional distributions and growth models on trees

• 成果类型: Article
• 署名作者: Liggett, TM
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1019160501
• 发表日期: 2000
• 页码: 1645-1665
• 关键词: catalan numbers; inequalities

摘要

We consider a sequence of probability measures v(n) obtained by conditioning 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 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 v(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.