Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T00:43:12.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Daniela Bertacchi  and
Fabio Zucca
Daniela Bertacchi*
Affiliation:
Università di Milano - Bicocca
Fabio Zucca*
Affiliation:
Politecnico di Milano
*
∗Postal address: Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, via Cozzi 53, 20125 Milano, Italy. Email address: [email protected]
∗∗Postal address: Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

Keywords

Branching random walkphase transitionmultigraphamenabilitytree

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 481 - 497
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Cassi, D. and Regina, S. (1992). Random walks on d-dimensional comb lattices. Modern Phys. Lett. B 6, 1397–1403.CrossRefGoogle Scholar
[2] Gerl, P. (1988). Random walks on graphs with a strong isoperimetric property. J. Theoret. Prob. 1, 171–187.Google Scholar
[3] Häggström, O. (2000). Markov random fields and percolation on general graphs. Adv. Appl. Prob. 32, 39–66.Google Scholar
[4] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
[5] Hueter, I. and Lalley, S. P. (2000). Anisotropic branching random walks on homogeneous trees. Prob. Theory Relat. Fields 116, 57–88.CrossRefGoogle Scholar
[6] Jagers, P. (1995). Branching processes as population dynamics. Bernoulli 1, 191–200.CrossRefGoogle Scholar
[7] Liggett, T. M. (1996). Branching random walks and contact processes on homogeneous trees. Prob. Theory Relat. Fields 106, 495–519.Google Scholar
[8] Liggett, T. M. (1999). Branching random walks on finite trees. In Perplexing Problems in Probability (Progress Prob. 44), Birkhäuser, Boston, MA, pp. 315–330.Google Scholar
[9] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys. 125, 337–353.CrossRefGoogle Scholar
[10] Lyons, R. (1990). Random walks and percolation on trees. Ann. Prob. 18, 931–958.Google Scholar
[11] Lyons, R. (2000). Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, 1099–1126.CrossRefGoogle Scholar
[12] Madras, N. and Schinazi, R. (1992). Branching random walks on trees. Stoch. Proc. Appl. 42, 255–267.CrossRefGoogle Scholar
[13] Pemantle, R. (1992). The contact process on trees. Ann. Prob. 20, 2089–2116.Google Scholar
[14] Pemantle, R. and Stacey, A. M. (2001). The branching random walk and contact process on Galton–Watson and nonhomogeneous trees. Ann. Prob. 29, 1563–1590.Google Scholar
[15] Stacey, A. M. (2003). Branching random walks on quasi-transitive graphs. Combin. Prob. Comput. 12, 345–358.CrossRefGoogle Scholar
[16] Woess, W. (2000). Random Walks on Infinite Graphs and Groups (Cambridge Tracts Math. 138). Cambridge University Press.CrossRefGoogle Scholar