# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] Aizenman, M, Geometric analysis of $$φ ^{4}$$ fields and Ising models. I, II, Commun. Math. Phys., 86, 1-48, (1982) · Zbl 0533.58034 [2] Aizenman, M; Barsky, DJ; Fernández, R, The phase transition in a general class of Ising-type models is sharp, J. Stat. Phys., 47, 343-374, (1987) [3] Aizenman, M; Fernández, R, On the critical behavior of the magnetization in high-dimensional Ising models, J. Stat. Phys., 44, 393-454, (1986) · Zbl 0629.60106 [4] Aizenman, M., Duminil-Copin, H., Sidoravicius, V.: Random currents and continuity of ising model’s spontaneous magnetization. Commun. Math. Phys. 334(2), 719-742 (2015) · Zbl 1315.82004 [5] Björnberg, J.E.: Vanishing critical magnetization in the quantum Ising model. Commun. Math. Phys. 337(2), 879-907 (2015) · Zbl 1321.82007 [6] Chelkak, D., Cimasoni, D., Kassel, A.: Revisiting the combinatorics of the 2D Ising model, 2015. to appear in Ann. Inst. Henri Poincaré Comb. Phys, Interact · Zbl 1380.82017 [7] Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2) 181(3), 1087-1138 (2015) · Zbl 1318.82006 [8] Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515-580 (2012) · Zbl 1257.82020 [9] Curtis, E.B., Ingerman, D., Morrow, J.A.: Circular planar graphs and resistor networks. Linear Algebra Appl. 283(1-3), 115-150 (1998) · Zbl 0931.05051 [10] Duminil-Copin, H.: Random currents expansion of the Ising model (2016). arXiv:1607.06933 · Zbl 1372.82017 [11] Fomin, S.: Loop-erased walks and total positivity. Trans. Am. Math. Soc. 353(9), 3563-3583 (2001) · Zbl 0973.15014 [12] Gantmacher, FR; Krein, MG, Sur LES matrices complètement non négatives et oscillatoires, fre, Compos. Math., 4, 445-476, (1937) · Zbl 0017.00102 [13] Gantmacher, F.R., Krein, M.G.: Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Wissenschaftliche Bearbeitung der deutschen Ausgabe: Alfred Stöhr. Mathematische Lehrbücher und Monographien, I. Abteilung, Bd. V, Akademie-Verlag, Berlin (1960) [14] Griffiths, R.B.: Correlations in ising ferromagnets. I, 1967. J. Math. Phys. 8(3), 478-483 [15] Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an ising ferromagnet in a positive external field, 1970. J. Math. Phys. 11(3), 790-795 (1970) [16] Groeneveld, J., Boel, R.J., Kasteleyn, P.W.: Correlation-function identities for general planar Ising systems. Phys. A: Stat. Mech. Appl. 93(1), 138-154 (1978) · Zbl 0048.45804 [17] Hongler, C.: Conformal Invariance of Ising Model Correlations, Ph.D. Thesis (2010) · Zbl 1304.82013 [18] Hongler, C., Smirnov, S.: The energy density in the planar Ising model. Acta Math. 211(2), 191-225 (2013) · Zbl 1287.82007 [19] Ising, E.: Beitrag zur Theorie des Ferromagnetismus, 1925FEB-APR, Z. Physik, 31, 253-258 [20] Kac, M; Ward, JC, A combinatorial solution of the two-dimensional Ising model, Phys. Rev., 88, 1332-1337, (1952) · Zbl 0048.45804 [21] Kager, W., Lis, M., Meester, R.: The signed loop approach to the ising model: foundations and critical point. J. Stat. Phys. 152(2), 353-387 (2013) · Zbl 1276.82009 [22] Karlin, Samuel, McGregor, James, coincidence probabilities, Pac. J. Math., 9, 1141-1164, (1959) · Zbl 0092.34503 [23] Kenyon, R.W., Wilson, D.B.: Combinatorics of tripartite boundary connections for trees and dimers. Electron J. Combinatorics [electronic only] 16(1) (2009), Research Paper R112, 28 pp · Zbl 1225.60020 [24] Kenyon, Richard W; Wilson, David B, Boundary partitions in trees and dimers, Trans. Am. Math. Soc., 363, 1325-1364, (2011) · Zbl 1230.60009 [25] Lawler, G.F.: A self-avoiding random walk. Duke Math. J. 47(3), 655-693 (1980) · Zbl 0445.60058 [26] Lis, M.: A short proof of the Kac-Ward formula. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3, 45-53 (2016) · Zbl 1331.05021 [27] Lupu, T., Werner, W.: A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electron. Commun. Probab. 21 (2016). 7 pp · Zbl 1338.60236 [28] Lusztig, G.: Total positivity in reductive groups, Lie theory and geometry, pp. 531-568 (1994) · Zbl 0845.20034 [29] Lusztig, G, Total positivity in partial flag manifolds, Represent. Theory, 2, 70-78, (1998) · Zbl 0895.14014 [30] Lusztig, G.: Introduction to total positivity. Positivity in Lie theory: open problems, pp. 133-145 (1998) · Zbl 0929.20035 [31] Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117-149 (1944) · Zbl 0060.46001 [32] Peierls, R, Ising’s, on, model of ferromagnetism, Proc. Cambridge Philos. Soc., 32, 477-481, (1936) · Zbl 0014.33604 [33] Postnikov, A.: Total positivity, Grassmannians, and networks (2006). arXiv:math/0609764 [34] Postnikov, A., Speyer, D., Williams, L.: Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Algebraic Combin. 30(2), 173-191 (2009) · Zbl 1264.20045 [35] Schoenberg, I.: Über variationsvermindernde lineare Transformationen. Mathematische Zeitschrift 32(1), 321-328 (1930) · JFM 56.0106.06 [36] Smirnov, S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2) 172(2), 1435-1467 (2010) · Zbl 1200.82011 [37] Talaska, K.: A formula for Plücker coordinates associated with a planar network. Int. Math. Res. Not. IMRN (2008), Art. ID rnn 081, 19 · Zbl 1170.05031 [38] Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. (2) 85, 808-816 (1952) · Zbl 0046.45304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.