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The planar Ising model and total positivity. (English) Zbl 1371.82024
A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). In the present paper, the author deals with total positivity in the planar Ising model. The author proves the positivity of determinants of matrices whose entries are, up to a sign, the boundary two-point spin correlation functions. The author obtains the total positivity of the Ising boundary two-point functions by representing them by means of alternating flows of K. Talaska which satisfy total positivity [Int. Math. Res. Not. 2008, Article ID rnn081, 19 p. (2008; Zbl 1170.05031)]. The author’s results on the Ising boundary two-point correlation functions are analogous to the results of S. Fomin [Trans. Am. Math. Soc. 353, No. 9, 3563–3583 (2001; Zbl 0973.15014)] on the walk matrices of random walks on planar graphs. The author gives an explicit combinatorial formula for the probability measure \(\overline{\mathbf{P}}^{A}_{\mathrm {d-curr}}\) on \(\Gamma_A\) induced from the double random current measure. The author computes the probability of \(\mathcal{P}_{A,B}\) under the double random current measure. The author also computes the probability of \(\omega\) induced from the random alternating flow measure. By using a classical Pfaffian formula of J. Groeneveld, R. J. Boel and P. W. Kasteleyn [Phys. A, Stat. Mech. Appl. 93, No. 1, 138–154 (1978)], the author presents a different proof involving the connection between double random currents and alternating flows. The author explains how the double random current measure is related to a measure on alternating flows on a finite connected planar graph \(G\) which is planar.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60C05 Combinatorial probability
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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