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Energy cascades as branching processes with emphasis on Neveu's approach to Derrida's random energy model

Part of: Markov processes

Published online by Cambridge University Press:  22 February 2016

Thierry Huillet
Thierry Huillet*
Affiliation:
Université de Cergy Pontoise and CNRS
*
∗ Postal address: Laboratoire de Physique Théorique et Modélisation, Université de Cergy Pontoise et CNRS (UMR8089), 95031, Neuville sur Oise, France. Email address: thierry.huillet@ptm.u-cergy.fr
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Abstract

Continuous-space-time branching processes (CSBP) are investigated in order to model random energy cascades. CSBPs are based on spectrally positive Lévy processes and, as such, are characterized by their corresponding Laplace exponents. Special emphasis is put on the CSBPs of Feller, Lamperti and Neveu and on their Poisson point process representations. The Neveu model (either supercritical or subcritical) is of particular interest in physics for its connection with the random energy model of Derrida, as revisited by Ruelle. Exploiting some connections between the partition functions of energy and the Poisson-Dirichlet distributions of Pitman and Yor, some information on the zero-temperature limit is extracted. Finally, for the subcritical versions of the three models, we compute the distribution of some of their interesting features: extinction time and probability, area under the profile (total energy) and width (maximal energy).

Keywords

Branching processenergy cascadePoisson point processLévy processPoisson-Dirichlet distribution

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 477 - 503
Copyright
Copyright © Applied Probability Trust 2003 

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