Abstract
We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the above expansion is compatible with the infinite differentiability of the free energy but does not imply its analyticity. The basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder.
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References
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
A. Berretti, Some properties of random Ising models. J. Statist. Phys. 38:483–496 (1985).
L. Bertini, E. N. M. Cirillo and E. Olivieri, Graded cluster expansion for lattice systems. Comm. Math. Phys. 258:405–443 (2005).
L. Bertini, E. N. M. Cirillo and E. Olivieri, A combinatorial proof of tree decay of semiinvariants. J. Stat. Phys. 115:395–413 (2004).
L. Bertini, E. N. M. Cirillo and E. Olivieri, Renormalization group in the uniqueness region: Weak Gibbsianity and convergence. Comm. Math. Phys. 261:323–378 (2006).
L. Bertini, E. N. M. Cirillo and E. Olivieri, in preparation.
R. L. Dobrushin, S. B. Shlosman, Constructive criterion for the uniqueness of Gibbs fields. Stat. Phys. Dyn. Syst., Birkhauser, 347–370 (1985).
R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions constructive description. J. Statist. Phys. 46:983–1014 (1987).
H. von Dreifus, A. Klein and J. F. Perez, Taming Griffiths’ singularities: Infinite differentiability of quenched correlation functions. Comm. Math. Phys. 170:21–39 (1995).
S. F. Edwards and P. W. Anderson, Theory of spin glasses. J. Phys. F Metal Phys. 5:965–974 (1975).
J. Fröhlich and J. Z. Imbrie, Improved perturbation expansion for disordered systems: Beating Griffiths’ singularities. Comm. Math. Phys. 96:145–180 (1984).
J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Second edition (Springer-Verlag, New York, 1987).
R. B. Griffiths, Non-analytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett. 23:17–19 (1969).
Y. Higuchi, Coexistence of infinite (*)-clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97:1–33 (1993).
J. L. Lebowitz and A. Martin-Löf, On the uniqueness of the equilibrium state for ising spin system. Comm. Math. Phys. 25:276–282 (1972).
F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I. The attractive case. Commun. Math. Phys. 161:447–486 (1994).
F. Martinelli, E. Olivieri and R. Schonniann, For 2-D lattice spin systems weak mixing implies strong mixing. Commun. Math. Phys. 165:33–47 (1994).
E. Olivieri, On a cluster expansion for lattice spin systems: A finite size condition for the convergence. J. Statist. Phys. 50:1179–1200 (1988).
E. Olivieri and P. Picco, Cluster expansion for D-dimensional lattice systems and finite volume factorization properties. J. Stat. Phys. 59:221–256 (1990).
L. Onsager, Crystal statistics I. A two-dimensional model with an order–disorder transition. Phys. Rev. 65:117–149 (1944).
D. Ruelle, On the use of small external fields in the problem of symmetry breakdown in statistical mechanics. Ann. Phys. 69:364–374 (1972).
A. Suto, Weak singularity and absence of metastability in random Ising ferromagnets. J. Phys. A 15:L7494–L752 (1982).
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MSC2000. Primary 82B44, 60K35.
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Bertini, L., Cirillo, E.N.M. & Olivieri, E. Perturbative Analysis of Disordered Ising Models Close to Criticality. J Stat Phys 126, 987–1006 (2007). https://doi.org/10.1007/s10955-006-9214-8
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DOI: https://doi.org/10.1007/s10955-006-9214-8