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Smirnov’s fermionic observable away from criticality. (English) Zbl 1339.60136
The authors provide new proofs of two results on the 2-dimensional Ising model on the square lattice. Let \(\sigma: \mathbb{Z}^2 \cap [-n, n]^2 \to \{-1,1\}\) be a function with \(\sigma_i \in \{-1,1\}\), \(i \in \mathbb{Z}^2 \cap [-n, n]^2\), where \(n\) is a large positive integer. The spin-spin correlation function of the 2-dimensional Ising model is defined as \[ \langle \sigma_x \sigma_y \rangle=\frac{\sum_\sigma e^{\beta \sum_{i,j} \sigma_i \sigma_j}\sigma_x \sigma_y}{\sum_\sigma e^{\beta \sum_{i,j} \sigma_i \sigma_j}}. \] The sum over \(i, j\) is over all nearest neighbor pairs \(i, j\). The sum over \(\sigma\) is over all functions \(\sigma\). First, they compute the critical inverse temperature \(\beta_c = \frac12 \ln(1 + \sqrt{2})\). Specifically, for \(\beta\) below \(\beta_c\), they show that the spin-spin correlation function decays exponentially fast as \(|x - y| \to \infty\). For \(\beta\) above \(\beta_c\), they show that the spin-spin correlation function is bounded away from zero as \(|x - y| \to \infty\). Second, they compute the exponential decay rate exactly for \(\beta\) below \(\beta_c\). Many years ago, Onsager and later M. Aizenman et al. [“The phase transition in a general class of Ising-type models is sharp”, J. Stat. Phys. 47, No. 3–4, 343–374 (1987; doi:10.1007/bf01007515)] computed the critical inverse temperature \(\beta_c\). B. M. McCoy and T. T. Wu [The two-dimensional Ising model. Cambridge, MA: Havard University Press (1973; Zbl 1094.82500)] computed the exponential decay rate exactly.
In both proofs, a crucial use of S. Smirnov’s fermionic observable [Ann. Math. (2) 172, No. 2, 1435–1467 (2010; Zbl 1200.82011)] is made. Many quantities, for example, the spin-spin correlation function, can be expressed in terms of the random cluster model. A graph consisting of vertices or sites \(\mathbb{Z}^2 \cap [-n, n]^2\) together with edges connecting nearest neighbor pairs is considered. They impose the Dobrushin boundary condition. With this boundary condition, every subgraph has a sequence of edges connecting a point on the boundary to another point on the boundary. Loosely speaking, this long sequence of edges forms an interface. A function whose domain is given by subgraphs and edges can be defined by the exponential of the purely imaginary number \(\frac{i}{2}\) times the total angular displacement of this interface from an edge on the boundary to an edge in the interior if the interface hits that edge and zero if the interface misses the edge. This function is evaluated with respect to the probability measure defined on the set of all subgraphs, and this is Smirnov’s fermionic observable. They show that Smirnov’s fermionic observable is massive harmonic, and they relate the gap between the eigenvalue 0 and the next eigenvalue to \(\beta\). They also prove a sort of “bosonization” result in Proposition 4.1 of their paper. One side of an equality is given in terms of the spin-spin correlation function, and the other side of the equality has the free field massive Green’s function.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B43 Percolation
82B27 Critical phenomena in equilibrium statistical mechanics
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