The Annals of Probability

Monotonicity of conditional distributions and growth models on trees

Thomas M. Liggett

Full-text: Open access


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.

Article information

Ann. Probab., Volume 28, Number 4 (2000), 1645-1665.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stochastic monotonicity contact process growth models on trees critical values critical exponents coupling Catalan numbers


Liggett, Thomas M. Monotonicity of conditional distributions and growth models on trees. Ann. Probab. 28 (2000), no. 4, 1645--1665. doi:10.1214/aop/1019160501.

Export citation


  • Efron, B. (1965). Increasing properties of P´olya frequency functions. Ann. Math. Statist. 36 272-279.
  • F ¨urlinger, J. and Hofbauer, J. (1985). q-Catalan numbers. J. Combin. Theory Ser. A 40 248-264.
  • Hilton, P. and Pedersen, J. (1991). Catalan numbers, their generalization and their uses. Math. Intelligencer 13 64-75.
  • Holley, R. (1974). Remarks on the FKG inequalities. Comm. Math. Phys. 36 227-231.
  • Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11 286-295.
  • Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities I: multivariate totally positive distributions. J. Multivariate Anal. 10 467- 498.
  • Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Springer, Berlin.
  • Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.
  • Pemantle, R. (2000). Towards a theory of negative dependence. Unpublished manuscript.
  • Preston, C. J. (1974). A generalization of the FKG inequalities. Comm. Math. Phys. 36 233-241.
  • Puha, A. L. (1999). A reversible nearest particle system on the homogeneous tree. J. Theoret. Probab. 12 217-254.
  • Puha, A. L. (2000). Critical exponents for a reversible nearest particle system on the binary tree. Ann. Probab. 28 395-415.
  • Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.
  • Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246-290.
  • Tretyakov, A. Y. and Konno, N. (2000). Survival probability for uniform model on binary tree: Critical behavior and scaling. In Soft Computing in Industrial Applications. Springer, New York.