Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-16T09:22:38.334Z Has data issue: false hasContentIssue false
  • English
  • Français

A branching process showing a phase transition

Published online by Cambridge University Press:  14 July 2016

T. Moore  and
J. L. Snell
T. Moore*
Affiliation:
Marietta College
J. L. Snell*
Affiliation:
Dartmouth College
*
∗Postal address: Marietta College, Marietta, Ohio 45750, U.S.A.
∗∗Postal address: Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that a phase transition for the Ising model on a tree graph can be detected by the limiting behavior of the magnetization. The relevant limit theorems are special cases of limit theorems for multitype branching processes.

Keywords

BRANCHING PROCESSESPHASE TRANSITIONLIMIT THEOREMSMARKOV RANDOM FIELDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 252 - 260
Copyright
Copyright © Applied Probability Trust 1979 

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.)

Footnotes

The preparation of this paper was supported in part by a grant from the National Science Foundation

References

Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Griffiths, R. B. (1964) Peierl's proof of spontaneous magnetization in two-dimensional Ising ferromagnet. Phys. Rev. A 136, 437–438.CrossRefGoogle Scholar
Ising, E. (1924) Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258.Google Scholar
Kesten, H. and Stigum, B. P. (1966) Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1463–1481.CrossRefGoogle Scholar
Malysev, V. A. (1975) The central limit theorem for Gibbsian random fields. Soviet Math. Dokl. 5, 41–45.Google Scholar
Matsuda, H. (1974) Infinite susceptibility without spontaneous magnetization. Progr. Theoret. Phys. 51, 1053–1063.CrossRefGoogle Scholar
Müller-Hartman, E. and Zittartz, J. (1974) New type of phase transition. Phys. Rev. Lett. 33, 893–897.Google Scholar
Peierls, R. E. (1936) On Ising's ferromagnet model. Proc. Camb. Phil. Soc. 32, 477–481.Google Scholar
Preston, C. J. (1974) Gibbs States on Countable Sets. Cambridge Tracts in Mathematics 68, Cambridge University Press.Google Scholar
Spitzer, F. (1975) Markov random fields on an infinite tree. Ann. Prob. 3, 387–398.Google Scholar