Robust phase transitions for Heisenberg and other models on general trees

Annals of Probability

Volumn 27, Issue 2, 1999, Pages 876-912

  Pemantle, Robin ^a   Steif, Jeffrey E ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Heisenberg models; Phase transitions; Symmetry breaking

Indexed keywords

EID: 0033456088     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1022677389     Document Type: Article

References (24)

1. ADEL'SON-VEL'SKII, G., VEISFEILER, B., LEMAN, A. and FARADZEV, I. (1969). Example of a graph without a transitive automorphism group. Soviet Math. Dokl. 10 440-441.
2. 2 Ising and Potts models, J. Statist. Phys. 50 1-40.
3. ASKEY, R. (1975). Orthogonal Polynomials and Special Functions. Arrowsmith, Bristol, England.
4. BROUWER, A., COHEN, A. and NEUMAIER, A. (1989). Distance Regular Graphs. Springer, New York.
5. BIGGS, N. (1993). Algebraic Graph Theory, 2nd ed. Cambridge Univ. Press.
6. CASSI, D. (1992). Phase transition and random walks on graphs: a generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures. Phys. Rev. Lett. 68 3631-3634.
7. EISELE, M. (1994). Phase transitions may be absent on graphs with transient random walks. Unpublished manuscript.
8. EVANS, W., KENYON, C., PERES, Y. and SCHULMAN, L. J. (1998). Broadcasting on trees and the Ising model. Ann. Appl. Probab. To appear.
9. FURSTENBERG, H. (1970). Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis. Symposium in Honor of Salomon Bochner (R. C. Gunning, ed.) 41-59. Princeton Univ. Press.
10. GEORGII, H.-O. (1988). Gibbs Measures and Phase Transitions, de Gruyter, New York.
11. HÄGGSTRÖM, O. (1996). The random-cluster model on a homogeneous tree. Probab. Theory Related Fields 104 231-253.
12. LEBOWITZ, J. L. and PENROSE, O. (1976). Thermodynamic limit of the free energy and correlation functions of spin systems. Acta Physica Austriaca, Suppl XVI 201-220.
13. LIGGETT, T. M. (1996). Multiple transition points for the contact process on a binary tree. Ann. Probab. 24 1675-1710.
14. LYONS, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-353.
15. LYONS, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
16. MERKL, F. and WAGNER, H. (1994). Recurrent random walks and the absence of continuous symmetry breaking on graphs. J. Statist. Phys. 75 153-165.
17. MONROE, J. L. and PEARCE, P. A. (1979). Correlation inequalities for vector spin models. J. Statist. Phys. 21 615-633.
18. NATTERER, F. (1986). The Mathematics of Computerized Tomography. Wiley, New York.
19. PATRASCIOIU A. and SEILER, E. (1992). Phase structure of two-dimensional spin models and percolation. J. Statist. Phys. 69 573-595.
20. PEMANTLE, R. (1992). The contact process on trees. Ann. Probab. 20 2089-2116.
21. PEMANTLE, R. and PERES, Y. (1999). Recursions on trees and the Ising model. Preprint.
22. RAINVILLE, E. D. (1960). Special Functions. MacMillan, New York.
23. STACEY, A. (1996). The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711-1726.
24. TERWILLIGER, P. (1998). Unpublished lecture notes.