Ising models, julia sets, and similarity of the maximal entropy measures
We study the phase transition of Ising models on diamondlike hierarchical lattices. Following an idea of Lee and Yang, one can make an analytic continuation of free energy of this model to the complex temperature plane. It is known that the Migdal-Kadanoff renormalization group of this model is a rational endomorphism (denoted byf) of Ĉ and that the singularities of the free energy lie on the Julia setJ(f). The aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onJ(f). Moreover, using this representation, we can show a close relationship between the critical exponent and local similarity of this measure.
Key WordsIsing models diamondlike hierarchical lattices renormalization groups Julia sets maximal entropy measures fractal structure
Unable to display preview. Download preview PDF.
- 1.T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions, II. Lattice gas and Ising model,Phys. Rev. 87:410–419 (1952).Google Scholar
- 2.A. A. Migdal, Recurrence equation in gauge field theory.Sov. Phys. JETP 69:810–822, 1457–1467 (1975).Google Scholar
- 3.L. P. Kadanoff, Notes on Migdal's recursion formulae,Ann. Phys. 100:359–394 (1976).Google Scholar
- 4.P. M. Bleher and E. Zalis, Asymptotics of the susceptibility for the Ising model on the hierarchical lattices,Commun. Math. Phys. 120:409–436 (1989).Google Scholar
- 5.P. M. Bleher and M. Yu. Lyubich, Julia sets and complex singularities in hierarchical Ising models.Commun. Math. Phys. 141:453–474 (1991).Google Scholar
- 6.B. Derrida, L. De Seze, and C. Itzykson, Fractal structure of zeros in hierarchical lattices,J. Stat. Phys. 30:559–570 (1983).Google Scholar
- 7.B. Derrida, L. De Seze, and J. M. Luck, Oscillatory critical amplitudes in hiearchical models,Commun. Math. Phys. 94:115–127 (1984).Google Scholar
- 8.B. Derrida, J.-P. Eckmann, and A. Erzan, Renormalization groups with periodic and aperiodic orbits,J. Phys. A: Math. Gen. 16:893–906 (1983).Google Scholar
- 9.H. Brolin, Invariant sets under iteration of rational functions,Ark. Mat. 6:103–144 (1965).Google Scholar
- 10.A. Freire, A. Lopes, and R. Mañé, An invariant measure for rational maps.Bol. Soc. Bras. Mat. 14:45–62 (1983).Google Scholar
- 11.R. Mañé, On the uniqueness of the maximizing measure for rational maps,Bol. Soc. Bras. Mat. 14:27–43 (1983).Google Scholar
- 12.M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,Ergodic Theory Dynam. Syst. 3:351–385 (1983).Google Scholar