Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-04-30T17:29:34.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Random fields on random graphs

Part of: Special processes Graph theory Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  01 July 2016

P. Whittle
P. Whittle*
Affiliation:
University of Cambridge
*
∗ Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

The distribution (1) used previously by the author to represent polymerisation of several types of unit also prescribes quite general statistics for a random field on a random graph. One has the integral expression (3) for its partition function, but the multiple complex form of the integral makes the nature of the expected saddlepoint evaluation in the thermodynamic limit unclear. It is shown in Section 4 that such an evaluation at a real positive saddlepoint holds, and subsidiary conditions narrowing down the choice of saddlepoint are deduced in Section 6. The analysis simplifies greatly in what is termed the semi-coupled case; see Sections 3, 5 and 7. In Section 8 the analysis is applied to an Ising model on a random graph of fixed degree r + 1. The Curie point of this model is found to agree with that deduced by Spitzer for an Ising model on an r-branching tree. This agreement strengthens the conclusion of ‘locally tree-like' behaviour of the graph, seen as an important property in a number of contexts.

MSC classification

Primary: 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C80: Random graphs
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 455 - 473
Copyright
Copyright © Applied Probability Trust 1992 

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.)

References

Aizenman, M. and Newman, C. M. (1984) Tree graph inequalities and critical behaviour in percolation models. J. Statist. Phys. 36, 107–143.Google Scholar
Derrida, B., Gardner, E. and Zippelius, A. (1987) An exactly solvable asymmetric neural network model. Europhysics Letters 4, 167–173.Google Scholar
Derrida, B. and Nadal, J. P. (1987) Learning and forgetting on asymmetric diluted neural networks. J. Stat. Phys. 23, 993–1011.Google Scholar
Derrida, B. and Pomeau, Y. (1986) Random networks of automata; a simple annealed approximation. Europhysics Letters 1, 45–49.Google Scholar
Derrida, B. and Weisbuch, G. (1986) Evolution of overlaps between configurations in random Boolean networks. J. de Physique (Paris) 47, 1297–1303.CrossRefGoogle Scholar
Flory, P. J. (1953) Principles of Polymer Chemistry, Cornell University Press.Google Scholar
Good, I. J. (1963) Cascade theory and the molecular weight averages of the sol fraction. Proc. Roy. Soc. A 272, 54–59.Google Scholar
Gordon, M. (1962) Good's theory of cascade processes applied to the statistics of polymer distributions. Proc. Roy. Soc. A 268, 240–259.Google Scholar
Grimmett, G. R. (1989) Percolation. Springer-Verlag, New York.Google Scholar
Hara, T. and Slade, G. (1988) Mean-field critical phenomena for percolation in high dimensions.Google Scholar
Hilhorst, H. J. and Nijmeijer, M. (1987) On the approach of the stationary state in Kauffman's random Boolean network. J. de Physique (Paris) 48, 185–191.Google Scholar
Kauffman, S. A. (1969) Random genetic nets. J. Theor. Biol. 22, 437–467.CrossRefGoogle Scholar
Kurten, K. E. (1988) Critical phenomena in model neural networks. Physics Letters A 129, 157–160.Google Scholar
Kurten, K. E. (1989) Dynamical phase transitions in short-ranged and long-ranged neural networks. J. Phys. France 50, 2313–2323.Google Scholar
Kurten, K. E. (1990) Dynamical learning in networks with sparse connectivity. In Parallel Processing in Neural Systems and Computers, ed. Eckmiller, R., Hartmann, G., and Hauske, G., Elsevier, Amsterdam 191–194.Google Scholar
Pittel, B., Woyczinski, W. A. and Mann, J. A. (1987) Random tree-type partitions as a model for acyclic polymerisation. Case Western Reserve University Preprint #87-79.Google Scholar
Spitzer, F. (1979) Markov random fields on an infinite tree. Ann. Prob. 3, 387–398.Google Scholar
Stockmayer, W. H. (1943) Theory of molecular size distribution and gel formation in branched chain polymers. J. Chem. Phys. 11, 45–55.Google Scholar
Whittle, P. (1965a) Statistical processes of aggregation and polymerisation. Proc. Camb. Phil. Soc. 61, 475–495.Google Scholar
Whittle, P. (1965b) The equilibrium statistics of a clustering process in the uncondensed phase. Proc. Roy. Soc. A 285, 501–519.Google Scholar
Whittle, P. (1980a, b, c) Polymerisation processes with intrapolymer bonding, I. II, III, Adv. Appl. Prob. 12, 94–115, 116-134, 135-153.CrossRefGoogle Scholar
Whittle, P. (1981) A direct derivation of the equilibrium distribution for a polymerisation process. Teoriya Veroyatnostei 26, 350–361.Google Scholar
Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, Chichester.Google Scholar
Whittle, P. (1989) The statistics of random directed graphs. J. Stat. Phys. 56, 499–516.Google Scholar
Whittle, P. (1990) Fields and flows on random graphs. In Disorder in Physical Systems, ed. Grimmett, G. R., and Welsh, D. J. A., pp. 337–348, Oxford University Press.Google Scholar