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Random systems in ultrametric spaces

Part of: Special processes Graph theory Combinatorial probability

Published online by Cambridge University Press:  01 February 2019

D. A. Dawson  and
L. G. Gorostiza
D. A. Dawson*
Affiliation:
Carleton University
L. G. Gorostiza*
Affiliation:
CINVESTAV
*
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada.
Department of Mathematics, CINVESTAV-IPN, Apartado Postal 14-740, Mexico DF 07000, Mexico. Email address: lgorosti@math.cinvestav.mx
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Abstract

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We discuss percolation and random walks in a class of homogeneous ultrametric spaces together with similarities and differences in ultrametric and Euclidean spaces. We briefly outline the role of these models in the study of interacting systems. Several open problems are presented.

Keywords

Ultrametric spacehierarchical latticerandom graphpercolationrandom walkinteracting systembranching

MSC classification

Primary: 05C80: Random graphs 05C81: Random walks on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60C05: Combinatorial probability
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 83 - 97
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Avetisov, V. A.,Bikulov, A. K. and Zubarev, A. P. (2014).Ultrametric random walk and dynamics of protein molecules.Proc. Steklov Inst. Math. 285,3–25.Google Scholar
[2]Berger, N. (2002).Transience, recurrence and critical behavior for long-range percolation.Commun. Math. Phys. 226,531–558.Google Scholar
[3]Bertacchi, D.,Lanchier, N. and Zucca, F. (2011).Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions.Ann. Appl. Prob. 21,1215–1252.Google Scholar
[4]Bojdecki, T.,Gorostiza, L. G. and Talarczyk, A. (2006).Limit theorems for occupation time fluctuations of branching systems. I. Long-range dependence.Stoch. Process. Appl. 116,1–18.Google Scholar
[5]Bojdecki, T.,Gorostiza, L. G. and Talarczyk, A. (2006).Limit theorems for occupation time fluctuations of branching systems. II. Critical and large dimensions.Stoch. Process. Appl. 116,19–35.Google Scholar
[6]Bojdecki, T.,Gorostiza, L. G. and Talarczyk, A. (2013).Oscillatory fractional Brownian motion.Acta Appl. Math. 127,193–215.Google Scholar
[7]Collet, P. and Eckmann, J.-P. (1978).A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics (Lecture Notes Phys. 74).Springer,Berlin.Google Scholar
[8]Crawford, N. and Sly, A. (2012).Simple random walk on long range percolation clusters I: heat kernel bounds.Prob. Theory Relat. Fields 154,753–786.Google Scholar
[9]Crawford, N. and Sly, A. (2013).Simple random walk on long-range percolation clusters II: scaling limits.Ann. Prob. 41,445–502.Google Scholar
[10]Dawson, D. A. and Gorostiza, L. G. (2007).Percolation in a hierarchical random graph.Commun. Stoch. Anal. 1,29–47.Google Scholar
[11]Dawson, D. A. and Gorostiza, L. G. (2013).Percolation in an ultrametric space.Electron. J. Prob. 18, 26pp.Google Scholar
[12]Dawson, D. A. and Gorostiza, L. G. (2018).Transience and recurrence of random walks on percolation clusters in an ultrametric space.J. Theoret. Prob. 31,494–526.Google Scholar
[13]Dawson, D. A. and Greven, A. (1993).Hierarchical models of interacting diffusions: multiple time scale phenomena, phase transition and pattern of cluster-formation.Prob. Theory Relat. Fields 96,435–473.Google Scholar
[14]Dawson, D. A.,Gorostiza, L. G. and Wakolbinger, A. (2001).Occupation time fluctuations in branching systems.J. Theoret. Prob. 14,729–796.Google Scholar
[15]Dawson, D. A.,Gorostiza, L. G. and Wakolbinger, A. (2004).Hierarchical equilibria of branching populations.Electron. J. Prob. 9,316–381.Google Scholar
[16]Dawson, D. A.,Gorostiza, L. G. and Wakolbinger, A. (2005).Degrees of transience and recurrence and hierarchical random walks.Potential Anal. 22,305–350.Google Scholar
[17]Dawson, D. A.,Greven, A. and Zähle, I. (2018). Continuous limits of multitype population models on the hierarchical group. In preparation.Google Scholar
[18]Dragovich, B.,Khrennikov, A. Y.,Kozyrev, S. V. and Volovich, I. V. (2009).On p-adic mathematical physics.𝑝-Adic Numbers Ultrametric Anal. Appl. 1,1–17.Google Scholar
[19]Evans, S. N. (1989).Local properties of Lévy processes on a totally disconnected group.J. Theoret. Prob. 2,209–259.Google Scholar
[20]Flatto, L. and Pitt, J. (1974).Recurrence criteria for random walks on countable Abelian groups.Illinois J. Math. 18,1–19.Google Scholar
[21]Greven, A.,Pfaffelhuber, P. and Winter, A. (2013).Tree-valued resampling dynamics martingale problems and applications.Prob. Theory Relat. Fields 155,789–838.Google Scholar
[22]Greven, A.,den Hollander, F.,Kliem, S. and Klimovsky, A. (2014).Renormalisation of hierarchically interacting Cannings processes.ALEA 11,43–140.Google Scholar
[23]Grimmett, G. R.,Kesten, H. and Zhang, Y. (1993).Random walk on the infinite cluster of the percolation model.Prob. Theory Relat. Fields 96,33–44.Google Scholar
[24]Janson, S.,Łuczak, T. and Rucinski, A. (2000).Random Graphs.Wiley-Interscience,New York.Google Scholar
[25]Koval, V.,Meester, R. and Trapman, P. (2012).Long-range percolation on the hierarchical lattice.Electron. J. Prob. 17, 21pp.Google Scholar
[26]Krishnapur, M. and Peres, Y. (2004).Recurrent graphs where two independent random walks collide finitely often.Electron. Commun. Prob. 9,72–81.Google Scholar
[27]Kumagai, T. (2014).Random Walks on Disordered Media and Their Scaling Limits (Lecture Notes Math. 2101).Springer,Cham.Google Scholar
[28]Liggett, T. M. (1974).A characterization of the invariant measures for an infinite particle system with interactions. II.Trans. Amer. Math. Soc. 198,201–213.Google Scholar
[29]Liggett, T. M. (1985).Interacting Particle Systems (Fundamental Principles Math. Sci. 276).Springer,New York.Google Scholar
[30]López-Mimbela, J. A. and Wakolbinger, A. (1997).Which critically branching populations persist? In Classical and Modern Branching Processes (IMA Vol. Mat. Appl. 84), eds K. B. Athreya and P. Jagers,Springer,New York, pp. 203–216.Google Scholar
[31]Lyons, T. (1983).A simple criterion for transience of a reversible Markov chain.Ann. Prob. 11,393–402.Google Scholar
[32]Lyons, R. and Peres, Y. (2016).Probability on Trees and Networks.Cambridge University Press.Google Scholar
[33]Panchenko, D. (2013).The Parisi ultrametricity conjecture.Ann. Math. 177,383–393.Google Scholar
[34]Rammal, R.,Toulouse, G. and Virasoro, M. A. (1986).Ultrametricity for physicists.Rev. Modern Phys. 58,765–788.Google Scholar
[35]Schikhof, W. H. (1984).Ultrametric Calculus: An Introduction to 𝑝-Adic Analysis.Cambridge University Press.Google Scholar
[36]Shang, Y. (2012).Percolation in a hierarchical lattice.Z. Naturforsch. 67,225–229.Google Scholar
[37]Spitzer, F. (1974).Recurrent random walk of an infinite particle system.Trans. Amer. Math. Soc. 198,191–199.Google Scholar
[38]Takeuchi, J. (1967).Moments of the last exit times.Proc. Japan Acad. 43,355–360.Google Scholar
[39]Vandembroucq, D. and Roux, S. (2004).Large-scale numerical simulations of ultrametric long-range depinning.Phys. Rev. E 70, 026103, 9pp.Google Scholar