The Annals of Probability

Smirnov’s fermionic observable away from criticality

V. Beffara and H. Duminil-Copin

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In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010) 1435–1467] defines an observable for the self-dual random-cluster model with cluster weight $q=2$ on the square lattice $\mathbb{Z} ^{2}$, and uses it to obtain conformal invariance in the scaling limit. We study this observable away from the self-dual point. From this, we obtain a new derivation of the fact that the self-dual and critical points coincide, which implies that the critical inverse temperature of the Ising model equals $\frac{1}{2}\log(1+\sqrt{2})$. Moreover, we relate the correlation length of the model to the large deviation behavior of a certain massive random walk (thus confirming an observation by Messikh [The surface tension near criticality of the 2d-Ising model (2006) Preprint]), which allows us to compute it explicitly.

Article information

Ann. Probab., Volume 40, Number 6 (2012), 2667-2689.

First available in Project Euclid: 26 October 2012

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Secondary: 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Ising model correlation length critical temperature massive harmonic function


Beffara, V.; Duminil-Copin, H. Smirnov’s fermionic observable away from criticality. Ann. Probab. 40 (2012), no. 6, 2667--2689. doi:10.1214/11-AOP689.

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  • [1] Aizenman, M., Barsky, D. J. and Fernández, R. (1987). The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47 343–374.
  • [2] Beffara, V. and Duminil-Copin, H. (2012). The self-dual point of the two-dimensional random-cluster model is critical for $q\geq1$. Probab. Theory Related Fields 153 511–542.
  • [3] Duminil-Copin, H., Hongler, C. and Nolin, P. (2012). Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 1165–1198.
  • [4] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
  • [5] Graham, B. T. and Grimmett, G. R. (2006). Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 1726–1745.
  • [6] Graham, B. T. and Grimmett, G. R. (2011). Sharp thresholds for the random-cluster and Ising models. Ann. Appl. Probab. 21 240–265.
  • [7] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [8] Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Z. Phys. 31 253–258.
  • [9] Kaufman, B. and Onsager, L. (1950). Crystal statistics. IV. Long-range order in a binary crystal. Unpublished manuscript.
  • [10] Kramers, H. A. and Wannier, G. H. (1941). Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60 252–262.
  • [11] Lenz, W. (1920). Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. Phys. Zeitschr. 21 613–615.
  • [12] McCoy, B. M. and Wu, T. T. (1973). The Two-Dimensional Ising Model. Harvard Univ. Press, Cambridge, MA.
  • [13] Messikh, R. J. (2006). The surface tension near criticality of the 2d-Ising model. Preprint. Available at arXiv:math/0610636.
  • [14] Peierls, R. (1936). On Ising’s model of ferromagnetism. Proc. Camb. Philos. Soc. 32 477–481.
  • [15] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467.