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Mean field frozen percolation. (English) Zbl 1181.82034
Summary: We define a modification of the Erdős-Rényi random graph process which can be regarded as the mean field frozen percolation process. We describe the behavior of the process using differential equations and investigate their solutions in order to show the self-organized critical and extremum properties of the critical frozen percolation model. We prove two limit theorems about the distribution of the size of the component of a typical frozen vertex.

82B43 Percolation
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
05C80 Random graphs (graph-theoretic aspects)
60F10 Large deviations
82B27 Critical phenomena in equilibrium statistical mechanics
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[1] Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5(1), 3–48 (1999) · Zbl 0930.60096 · doi:10.2307/3318611
[2] Aldous, D.J.: The percolation process on a tree where infinite clusters are frozen. Math. Proc. Camb. Philos. Soc. 128(3), 465–477 (2000) · Zbl 0961.60096 · doi:10.1017/S0305004199004326
[3] Drossel, B., Schwabl, F.: Self-organized critical forest-fire model. Phys. Rev. Lett. 69(11), 1629–1632 (1992) · doi:10.1103/PhysRevLett.69.1629
[4] Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971) · Zbl 0219.60003
[5] Lushnikov, A.: Some new aspects of coagulation theory. Izv. Akad. Nauk. SSSR, Ser. Fiz. Atmos. I Okeana 14(10), 738–743 (1978)
[6] Ráth, B., Tóth, B.: Erdos-Rényi random graphs + forest fires = self-organized criticality. Electron. J. Probab. 14, 1290–1327 (2009)
[7] van den Berg, J., Tóth, B.: A signal-recovery system: asymptotic properties, and construction of an infinite-volume process. Stoch. Process. Appl. 96(2), 177–190 (2001) · Zbl 1058.60093 · doi:10.1016/S0304-4149(01)00113-2
[8] Ziff, R.M., Ernst, M.H., Hendriks, E.M.: Kinetics of gelation and universality. J. Phys. A, Math. Gen. 16, 2293–2320 (1983) · doi:10.1088/0305-4470/16/10/026
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