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Mean Field Frozen Percolation

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Abstract

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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Drossel, B., Schwabl, F.: Self-organized critical forest-fire model. Phys. Rev. Lett. 69(11), 1629–1632 (1992)

    Article  ADS  Google Scholar 

  4. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    MATH  Google Scholar 

  5. Lushnikov, A.: Some new aspects of coagulation theory. Izv. Akad. Nauk. SSSR, Ser. Fiz. Atmos. I Okeana 14(10), 738–743 (1978)

    Google Scholar 

  6. Ráth, B., Tóth, B.: Erdős-Rényi random graphs + forest fires = self-organized criticality. Electron. J. Probab. 14, 1290–1327 (2009)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  8. Ziff, R.M., Ernst, M.H., Hendriks, E.M.: Kinetics of gelation and universality. J. Phys. A, Math. Gen. 16, 2293–2320 (1983)

    Article  MathSciNet  ADS  Google Scholar 

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Correspondence to Balázs Ráth.

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Ráth, B. Mean Field Frozen Percolation. J Stat Phys 137, 459–499 (2009). https://doi.org/10.1007/s10955-009-9863-5

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  • DOI: https://doi.org/10.1007/s10955-009-9863-5

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