Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T12:16:00.209Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean field theory for probabilists. Bernoulli 5, 3–48.CrossRefGoogle Scholar
[2] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
[3] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[4] Bertoin, J. (2000). Subordinators, Lévy Processes with No Negative Jumps and Branching Processes (Lecture Notes Concentrated Adv. Course Lévy Process. 8). MaPhySto, University of Aarhus.Google Scholar
[5] Bertoin, J. and Le Gall, J. F. (2000). The Bolthausen–Sznitman coalescent and the genealogy of continuous state branching processes. Prob. Theory Relat. Fields 117, 249–266.CrossRefGoogle Scholar
[6] Bingham, N. H. (1976). Continuous branching processes and spectral positivity. Stoch. Process. Appl. 4, 217–242.Google Scholar
[7] Carlton, M. (1999). Applications of the two-parameter Poisson–Dirichlet distribution. Doctoral Thesis, University of California, Los Angeles.Google Scholar
[8] Darling, D. A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73, 95–107.CrossRefGoogle Scholar
[9] Derrida, B. (1981). Random energy model: an exactly solvable model of disordered systems. Phys. Rev. B. 24, 2613–2626.Google Scholar
[10] Derrida, B. (1987). Statistical properties of randomly broken objects and of multivalley structures in disordered systems. J. Phys. A 20, 5273–5288.Google Scholar
[11] Devroye, L. (1990). A note on Linnik's distribution. Statist. Prob. Lett. 9, 305–306.Google Scholar
[12] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, New York.Google Scholar
[13] Engen, S. (1978). Stochastic Abundance Models. Chapman and Hall, London.Google Scholar
[14] Ewens, W. J. (1988). Population genetics theory—the past and the future. In Mathematical and Statistical Problems in Evolution, ed. Lessard, S., University of Montreal Press, pp. 117–227.Google Scholar
[15] Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
[16] Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Prob. 11, 669–677.CrossRefGoogle Scholar
[17] Grey, D. R. (1977). Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702–716.Google Scholar
[18] Huillet, T. and Ben Alaya, M. (2000). On continuous-state branching process models for random energy cascades. Submitted.Google Scholar
[19] Jirina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8, 292–313.CrossRefGoogle Scholar
[20] Kingman, J. F. C. (1975). Random discrete distributions. J. R. Statist. Soc. B 37, 1–22.Google Scholar
[21] Kozubowski, T. J. and Rachev, S. T. (1999). Univariate geometric stable laws. J. Comput. Anal. Appl. 1, 177–217.Google Scholar
[22] Kozubowski, T. J., Podgorski, K. and Samorodnitsky, G. (1998). Tails of Lévy measure of geometric stable random variables. Extremes 1, 367–378.Google Scholar
[23] Lamperti, J. W. (1967). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.Google Scholar
[24] Le Gall, J. F. Random Trees and Spatial Branching Processes (Lecture Notes Concentrated Adv. Course Lévy Process. 9). MaPhySto, University of Aarhus.Google Scholar
[25] Lévy, P., (1937). Théorie de l'Addition des Variables Aléatoires. Gauthier Villars, Paris.Google Scholar
[26] Lukacs, E. (1983). Developments in Characteristic Function Theory. Griffin, London.Google Scholar
[27] Neveu, J. (1992). A continuous state branching process in relation with the GREM model of spin glass theory. Tech. Rep. 267, Ecole Polytechnique, Palaiseau.Google Scholar
[28] Perman, M. (1993). Order statistics for jumps of normalized subordinators. Stoch. Process. Appl. 46, 267–281.CrossRefGoogle Scholar
[29] Pillai, R. N. (1990). On Mittag–Leffler functions and related distributions. Ann. Inst. Statist. Math. 42, 157–161.Google Scholar
[30] Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. Appl. Prob. 28, 525–539.Google Scholar
[31] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.CrossRefGoogle Scholar
[32] Pitman, J. and Yor, M. (1997). The two parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
[33] Ruelle, D. (1987). Mathematical reformulation of Derrida's REM and GREM. Commun. Math. Phys. 108, 225–239.Google Scholar
[34] Schuh, H. J. and Barbour, A. D. (1977). On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.Google Scholar
[35] Sevast´janov, B. A. (1968). Branching processes. Mat. Zametki 4, 239–251.Google Scholar
[36] Silverstein, M. L. (1968). A new approach to local times, J. Math. Mech. 17, 1023–1054.Google Scholar
[37] Spitzer, F. (1976). Principles of Random Walks (Graduate Texts Math. 34), 2nd edn. Springer, New York.Google Scholar
[38] Takacs, L. (1966). Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar