We establish a precise connection between gelation of polymers in Lushnikov's model and the emergence of the giant component in random graph theory. This is achieved by defining a modified version of the Erdös-Rényi process; when contracting to a polymer state space, this process becomes a discrete-time Markov chain embedded in Lushnikov's process. The asymptotic distribution of the number of transitions in Lushnikov's model is studied. A criterion for a general Markov chain to retain the Markov property under the grouping of states is derived. We obtain a noncombinatorial proof of a theorem of Erdös-Rényi type.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
M. V. Smoluchovski, Versuch einer mathematischen Theorie der Koagulationskinetic kolloider Lösunger,Z. Phys. Chem. 92:129–168 (1917).
R. L. Drake, A general mathematical survey of the coagulation equation, inTopics in Current Aerosol Research, part 2, G. M. Hidy and J. R. Brock, eds. (Pergamon, Oxford, 1972).
F. Family and D. P. Landau (eds.),Kinetics of Aggregation and Gelation (North-Holland, Amsterdam, 1984).
W. H. White, A global existence theorem for Smoluchovski's coagulation equations,Proc. Am. Math. Soc. 80:273–276 (1980).
A. H. Marcus, Stochastic coalescence,Technometrics 10:133–143 (1968).
A. A. Lushnikov, Certain new aspects of the coagulation theory,Izv. Atm. Ok. Fiz. 14:738–743 (1978).
E. M. Hendriks, J. L. Spouge, M. Eibl, and M. Schrekenberg, Exact solutions for random coagulation processes,Z. Phys. B 58:219–228 (1985).
P. Van Dongen, Fluctuations in coagulating systems,J. Stat. Phys. 49:879–926 (1987).
R. M. Ziff, M. H. Ernst, and E. M. Hendriks, Kinetics of gelation and universality,J. Phys. A 16:2293–2320 (1983).
E. Buffet and J. V. Pulé, On Lushnikov's model of gelation,J. Stat. Phys. 58:1041–1058 (1990).
P. Erdös and A. Rényi, On random graphs I,Publ. Math. Debrecen. 6:290–297 (1959).
P. Erdös and A. Rényi, On the evolution of random graphs,Publ. Math. Inst. Hung. Acad. Sci. 5:17–61 (1960).
B. Bollobas,Random Graphs (Academic Press, London, 1985).
E. M. Palmer,Graphical Evolution (Wiley, New York, 1985).
About this article
Cite this article
Buffet, E., Pulé, J.V. Polymers and random graphs. J Stat Phys 64, 87–110 (1991). https://doi.org/10.1007/BF01057869
- Gelation of polymers
- giant component of random graph
- grouping of states in a Markov chain
- Erdös-Rényi theorem