Hostname: page-component-5db58dd55d-m58mf Total loading time: 0 Render date: 2026-06-09T13:09:51.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating Critical Parameters of Branching Random Walks

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: daniela.bertacchi@unimib.it
∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: fabio.zucca@polimi.it
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.

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

Keywords

Branching random walkcritical parameterspercolationgraphs

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 463 - 478
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45, 481–497.Google Scholar
[2] Bertacchi, D. and Zucca, F. (2009). Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes. J. Statist. Phys. 134, 53–65 Google Scholar
[3] Bertacchi, D., Posta, G. and Zucca, F. (2007). Ecological equilibrium for restrained random walks. Ann. Appl. Prob. 17, 1117–1137.Google Scholar
[4] Bramson, M. and Durrett, R. (1988). A simple proof of the stability criterion of Gray and Griffeath. Prob. Theory Relat. Fields 80, 293–298.Google Scholar
[5] Coulhon, T., Grigor'yan, A. and Zucca, F. (2005). The discrete integral maximum principle and its applications. Tohoku Math. J. 57, 559–587.CrossRefGoogle Scholar
[6] Durrett, R. (1995). Ten Lectures on Particle Systems (Lectures Notes Math. 1608). Springer, Berlin.Google Scholar
[7] Durrett, R. and Neuhauser, C. (1991). Epidemics with recovery in D=2. Ann. Appl. Prob. 1, 189–206.CrossRefGoogle Scholar
[8] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
[9] Grimmett, G. (1999). Percolation. Springer, Berlin.CrossRefGoogle Scholar
[10] Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20.Google Scholar
[11] Hueter, I. and Lalley, S. P. (2000). {Anisotropic branching random walks on homogeneous trees}. Prob. Theory Relat. Fields 116, 57–88.Google Scholar
[12] Liggett, T. M. (1996). {Branching random walks and contact processes on homogeneous trees}. Prob. Theory Relat. Fields 106, 495–519.Google Scholar
[13] 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
[14] Liggett, T. M. and Spitzer, F. (1981). Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitsth. 56, 443–468.CrossRefGoogle Scholar
[15] Lyons, R. (2000). Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, 1099–1126.CrossRefGoogle Scholar
[16] Madras, N. and Schinazi, R. (1992). Branching random walks on trees. Stoch. Process. Appl. 42, 255–267.CrossRefGoogle Scholar
[17] Mountford, T. and Schinazi, R. (2005). A note on branching random walks on finite sets. J. Appl. Prob. 42, 287–294.CrossRefGoogle Scholar
[18] Neuhauser, C. (1992). Ergodic theorems for the multitype contact process. Prob. Theory Relat. Fields 91, 467–506.Google Scholar
[19] 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.CrossRefGoogle Scholar
[20] Seneta, E. (2006). Non-Negative Matrices and Markov Chains. Springer, New York.Google Scholar
[21] Schinazi, R. (2003). On the role of social clusters in the transmission of infectious diseases. J. Theoret. Biol. 225, 59–63.Google Scholar
[22] Schinazi, R. (2005). Mass extinctions: an alternative to the Allee effects. Ann. Appl. Prob. 15, 984–991.Google Scholar
[23] Stacey, A. M. (2003). Branching random walks on quasi-transitive graphs. Combinatorics Prob. Comput. 12, 345–358.CrossRefGoogle Scholar
[24] Van Den Berg, J., Grimmett, G. R. and Schinazi, R. B. (1998). Dependent random graphs and spatial epidemics. Ann. Appl. Prob. 8, 317–336.Google Scholar
[25] Woess, W. (2000). Random Walks on Infinite Graphs and Groups (Camb. Tracts Math. 138). Cambridge University Press.CrossRefGoogle Scholar