The Annals of Probability

Brownian excursions, critical random graphs and the multiplicative coalescent

David Aldous

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Abstract

Let $(B^t (s), 0 \leq s < \infty)$ be reflecting inhomogeneous Brownian motion with drift $t - s$ at time $s$, started with $B^t (0) = 0$. Consider the random graph $\mathscr{G}(n, n^{-1} + tn^{-4/3})$, whose largest components have size of order $n^{2/3}$. Normalizing by $n^{-2/3}$, the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of $B^t$ (Corollary 2). The dynamics of merging of components as $t$ increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors $\mathsf{x}$ of nonnegative real cluster sizes $(x_i)$, and clusters with sizes $x_i$ and $x_j$ merge at rate $x_i x_j$. The multiplicative coalescent is shown to be a Feller process on l_2$. The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time $-\infty$; the existence of such a process is not obvious.

Article information

Source
Ann. Probab. Volume 25, Number 2 (1997), 812-854.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1024404421

Mathematical Reviews number (MathSciNet)
MR1434128

Digital Object Identifier
doi:10.1214/aop/1024404421

Zentralblatt MATH identifier
0877.60010

Subjects
Primary: 60C05: Combinatorial probability 60J50: Boundary theory

Keywords
Brownian motion Brownian excursion Markov process random graph critical point stochastic coalescent stochastic coagulation weak convergence

Citation

Aldous, David. Brownian excursions, critical random graphs and the multiplicative coalescent. The Annals of Probability 25 (1997), no. 2, 812--854. doi:10.1214/aop/1024404421. http://projecteuclid.org/euclid.aop/1024404421.


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References

  • [1] Aldous, D. J. (1991). The continuum random tree. II: an overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23-70. Cambridge Univ. Press.
  • [2] Aldous, D. J. (1993). The continuum random tree. III. Ann. Probab. 21 248-289.
  • [3] Aldous, D. J. (1996). Deterministic and stochastic models for coalescence: a review of the mean-field theory for probabilists. Unpublished manuscript.
  • [4] Aldous, D. J. and Limic, V. (1996). The entrance boundary of the multiplicative coalescent. Unpublished manuscript.
  • [5] Aldous, D. J. and Pitman, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures and Algorithms 5 487-512.
  • [6] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press.
  • [7] Bollob´as, B. (1985). Random Graphs. Academic Press, London.
  • [8] Borgs, C., Chayes, J., Kesten, H. and Spencer, J. (1996). The birth of the infinite cluster: finite-size scaling in percolation. Unpublished manuscript.
  • [9] Donnelly, P. and Joyce, P. (1992). Weak convergence of population genealogical processes to the coalescent with ages. Ann. Probab. 20 322-341.
  • [10] Drake, R. L. (1972). A general mathematical survey of the coagulation equation. International Reviews in Aerosol Physics and Chemistry 3 201-376. Pergamon, Elmsford, NY.
  • [11] Erd os, P. and R´enyi, A. (1960). On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 17-61.
  • [12] Erd os, P. and R´enyi, A. (1961). On the evolution of random graphs. Bull. Inst. Internat. Statist. 38 343-347.
  • [13] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [14] Evans, S. N. and Pitman, J. (1996). Construction of Markovian coalescents. Technical Report 465, Dept. Statistics, Univ. California, Berkeley.
  • [15] Janson, S. (1993). Multicyclic components in a random graph process. Random Structures and Algorithms 4 71-84.
  • [16] Janson, S., Knuth, D. E., Luczak, T. and Pittel, B. (1993). The birth of the giant component. Random Structures and Algorithms 4 233-358.
  • [17] Kallenberg, O. (1983). Random Measures. Akademie-Verlag, Berlin.
  • [18] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248.
  • [19] Luczak, T., Pittel, B. and Wierman, J. C. (1994). The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341 721-748.
  • [20] Lushnikov, A. A. (1978). Coagulation in finite systems. J. Colloid and Interface Science 65 276-285.
  • [21] Marcus, A. H. (1968). Stochastic coalescence. Technometrics 10 133-143.
  • [22] Martin-L ¨of, A. (1996). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. To appear.
  • [23] Pitman, J. (1992). Partition structures derived from Brownian motion and stable subordinators. Technical Report 346, Dept. Statistics, Univ. California, Berkeley.
  • [24] Pitman, J. W. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900.
  • [25] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [26] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales: It o Calculus 2. Wiley, New York.
  • [27] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales: Foundations 1, 2nd ed. Wiley, New York.
  • [28] Spencer, J. (1996). Enumerating graphs and Brownian motion. Technical report, Courant Institute, New York.
  • [29] Tak acs, L. (1991). A Bernouilli excursion and its various applications. Adv. in Appl. Probab. 23 557-585.
  • [30] van Dongen, P. G. J. (1987). Fluctuations in coagulating systems. II. J. Statist. Phys. 49 927-975.
  • [31] van Dongen, P. G. J. and Ernst, M. H. (1987). Fluctuations in coagulating systems. J. Statist. Phys. 49 879-926.