The Annals of Probability

The critical Ising model on trees, concave recursions and nonlinear capacity

Robin Pemantle and Yuval Peres

Full-text: Open access


We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L3 capacity arises. In particular, on a spherically symmetric tree that has nαbn vertices at level n (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if α≤1/2. Our proofs are based on a new link between nonlinear recursions on trees and Lp capacities.

Article information

Ann. Probab., Volume 38, Number 1 (2010), 184-206.

First available in Project Euclid: 25 January 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 31C45: Other generalizations (nonlinear potential theory, etc.)

Ising model reconstruction capacity nonlinear potential theory trees iteration spin-glass recursion


Pemantle, Robin; Peres, Yuval. The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38 (2010), no. 1, 184--206. doi:10.1214/09-AOP482.

Export citation


  • Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields 131 311–340.
  • Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79 473–482.
  • Borgs, C., Chayes, J., Mossel, E. and Roch, S. (2006). The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In IEEE Foundations of Computer Science (FOCS) 518–530. FOCS, Los Alamitos, CA.
  • Chayes, J. T., Chayes, L., Sethna, J. P. and Thouless, D. J. (1986). A mean field spin glass with short-range interactions. Comm. Math. Phys. 106 41–89.
  • Ding, J., Lubetzky, E. and Peres, Y. (2009). The mixing time evolution of Glauber dynamics for the mean-field Ising model. Comm. Math. Phys. 289 725–764.
  • Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410–433.
  • Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics 9. De Gruyter, Berlin.
  • Häggström, O. (1996). The random-cluster model on a homogeneous tree. Probab. Theory Related Fields 104 231–253.
  • Ioffe, D. (1996a). On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137–143.
  • Ioffe, D. (1996b). Extremality of the disordered state for the Ising model on general trees. In Trees (Versailles, 1995). Progress in Probability 40 3–14. Birkhäuser, Basel.
  • Janson, S. and Mossel, E. (2004). Robust reconstruction on trees is determined by the second eigenvalue. Ann. Probab. 32 2630–2649.
  • Kenyon, C., Mossel, E. and Peres, Y. (2001). Glauber dynamics on trees and hyperbolic graphs. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001) 568–578. IEEE Computer Soc., Los Alamitos, CA.
  • Martinelli, F., Sinclair, A. and Weitz, D. (2004). Glauber dynamics on trees: Boundary conditions and mixing time. Comm. Math. Phys. 250 301–334.
  • Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337–353.
  • Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.
  • Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20 2043–2088.
  • Marchal, P. (1998). The best bounds in a theorem of Russell Lyons. Electron. Comm. Probab. 3 91–94.
  • Mossel, E. (2004). Survey: Information flow on trees. In Graphs, Morphisms and Statistical Physics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 63 155–170. Amer. Math. Soc., Providence, RI.
  • Mossel, E. and Peres, Y. (2003). Information flow on trees. Ann. Appl. Probab. 13 817–844.
  • Murakami, A. and Yamasaki, M. (1992). Nonlinear potentials on an infinite network. Mem. Fac. Sci. Shimane Univ. 26 15–28.
  • Pemantle, R. and Steif, J. E. (1999). Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 876–912.
  • Preston, C. J. (1974). Gibbs States on Countable Sets. Cambridge Univ. Press, London.
  • Preston, C. (1976). Random Fields. Lecture Notes in Math. 534. Springer, Berlin.
  • Sly, A. (2009). Reconstruction for the potts model. In Forty first ACM Symposium on Theory of Computing (STOC), 581–590.
  • Soardi, P. M. (1993). Morphisms and currents in infinite nonlinear resistive networks. Potential Anal. 2 315–347.
  • Soardi, P. M. (1994). Potential Theory on Infinite Networks. Lecture Notes in Math. 1590. Springer, Berlin.