Hostname: page-component-5db58dd55d-8mwbx Total loading time: 0 Render date: 2026-06-01T15:04:27.999Z Has data issue: false hasContentIssue false
  • English
  • Français

A Point Process Describing the Component Sizes in the Critical Window of the Random Graph Evolution

Published online by Cambridge University Press:  01 July 2007

SVANTE JANSON  and
JOEL SPENCER
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden (e-mail: svante.janson@math.uu.se, http://www.math.uu.se/~svante/)
JOEL SPENCER
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA (e-mail: spencer@cs.nyu.edu, http://www.cs.nyu.edu/cs/faculty/spencer/)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a point process describing the asymptotic behaviour of sizes of the largest components of the random graph G(n, p) in the critical window, that is, for p = n−1 + λn−4/3, where λ is a fixed real number. In particular, we show that this point process has a surprising rigidity. Fluctuations in the large values will be balanced by opposite fluctuations in the small values such that the sum of the values larger than a small ϵ (a scaled version of the number of vertices in components of size greater than εn2/3) is almost constant.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 4 , July 2007 , pp. 631 - 658
Copyright
Copyright © Cambridge University Press 2007

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.)

Article purchase

Temporarily unavailable

References

[1]Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.CrossRefGoogle Scholar
[2]Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation, Oxford University Press, Oxford.CrossRefGoogle Scholar
[3]Billingsley, P. (1968) Convergence of Probability Measures, Wiley, New York.Google Scholar
[4]Bollobás, B. (1985) Random Graphs, Academic Press, London.Google Scholar
[5]Chassaing, P. and Janson, S. (2004) The center of mass of the ISE and the Wiener index of trees. Electron. Comm. Probab., 9 # 20, 178–187.Google Scholar
[6]Gut, A. (2005) Probability: A Graduate Course, Springer, New York.Google Scholar
[7]Janson, S. (1993) Multicyclic components in a random graph process. RSA 4 71–84.Google Scholar
[8]Janson, S. (1997) Gaussian Hilbert Spaces, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[9]Janson, S. (2003) Cycles and unicyclic components in random graphs. CPC 12 27–52.Google Scholar
[10]Janson, S. (2003) The Wiener index of simply generated random trees. RSA 22 337–358.Google Scholar
[11]Janson, S., Knuth, D. E., Łuczak, T. and Pittel, B. (1993) The birth of the giant component. RSA 3 233–358.Google Scholar
[12]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley, New York.CrossRefGoogle Scholar
[13]Kallenberg, O. (1983) Point Processes, Akademie-Verlag, Berlin.Google Scholar
[14]Kallenberg, O. (2002) Foundations of Modern Probability, 2nd edn, Springer, New York.CrossRefGoogle Scholar
[15]Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes, Springer, New York.CrossRefGoogle Scholar
[16]Louchard, G. (1984) Kac's formula, Lévy's local time and Brownian excursion. J. Appl. Probab. 21 479–499.CrossRefGoogle Scholar
[17]Louchard, G. (1984) The Brownian excursion area: A numerical analysis. Comput. Math. Appl. 10 413–417. Erratum: Comput. Math. Appl. Part A 12 (1986) 375.CrossRefGoogle Scholar
[18]Łuczak, T., Pittel, B. and Wierman, J. C. (1994) The structure of a random graph near the point of the phase transition. Trans. Amer. Math. Soc. 341 721–748.CrossRefGoogle Scholar
[19]Otter, R. (1949) The multiplicative process. Ann. Math. Statist. 20 206–224.CrossRefGoogle Scholar
[20]Peres, Y. (2005) Personal communication.Google Scholar
[21]Pitman, J. (1998) Enumerations of trees and forests related to branching processes and random walks. In Microsurveys in Discrete Probability: Princeton, NJ, 1997, Vol. 41 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., AMS, Providence, RI, pp. 163–180.CrossRefGoogle Scholar
[22]Pittel, B. (1990) On tree census and the giant component in sparse random graphs. RSA 1 311–342.Google Scholar
[23]Pittel, B. and Wormald, N. C. (2005) Counting connected graphs inside-out. J. Combin. Theory Ser. B 93 127–172.CrossRefGoogle Scholar
[24]Spencer, J. (1997) Enumerating graphs and Brownian motion. Comm. Pure Appl. Math. 50 291–294.3.0.CO;2-6>CrossRefGoogle Scholar
[25]Wright, E. M. (1977) The number of connected sparsely edged graphs. J. Graph Theory 1 317–330.CrossRefGoogle Scholar