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Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results. (English) Zbl 1292.47016

Let \(\varphi\) be an integrable function on the unit circle with the Fourier coefficients \(\varphi_k\). The Toeplitz matrices \(T_n(\varphi)\) and Toeplitz determinants \(D_n(\varphi)\) with symbol \(\varphi\) are given by \(T_n(\varphi)=\{\varphi_{j-k}\}_{0\leq j,k\leq n-1}\) and \(D_n(\varphi)=\det T_n(\varphi)\), respectively. The authors say that “our goal in this paper is not to review the development and applications in a comprehensive way, but rather to focus on one central problem in the theory of Toeplitz determinants, the Szegő strong limit theorem”. This theorem says that, if \(\varphi\) is a positive function in \(C^{1+\varepsilon}\) with \(\varepsilon>0\), then \[ \lim\limits_{n\to\infty}{D_n(\varphi)\over e^{n(\log\varphi)_0}}=e^{E(\varphi)}, \] where \(E(\varphi)=\sum_{k=1}^\infty k|(\log\varphi)_k|^2\).
In the survey paper under review, the history of the Szegő strong theorem and its further generalizations is discussed in detail in connection with questions raised in the analysis of the two-dimensional Ising model, one of the central models in statistical mechanics. This model concerns the interaction of random spins \(\sigma_{i,j} =\pm 1\) at sites \((i,j)\) in the lattice \({\mathbb Z}^2\). The situation of the so-called ferromagnetic system, i.e., when parallel spins have lower energy than antiparallel spins, is of great interest. This happens, in particular, when only nearest-neighbor spins interact and the interaction energy is given by \(-J_1\sigma_{i,j}\sigma_{i,j+1}-J_2\sigma_{i,j}\sigma_{i+1,j}\), where the vertical and horizontal interaction constants, \(J_1\) and \(J_2\) respectively, are translation invariant and positive.
The paper is nicely written and is addressed to a wide audience. Specialists in many areas will enjoy an abundance of factual material and many interesting historical details. The list of references is up to date, it contains 199 items.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15B05 Toeplitz, Cauchy, and related matrices
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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References:

[1] Ablowitz, Solitons, nonlinear evolution equations and inverse scattering (1991) · Zbl 0762.35001
[2] Ablowitz, Asymptotic solutions of the Korteweg-deVries equation, Studies in Appl. Math. 57 (1) pp 13– (1976/77) · Zbl 0369.35055
[3] Au-Yang, Wavevector-dependent susceptibility in quasiperiodic Ising models. Proceedings of the Baxter Revolution in Mathematical Physics (Canberra, 2000), J. Statist. Phys. 102 (3-4) pp 501– (2001) · Zbl 0999.82015
[4] Au-Yang, Ising correlations at the critical temperature, Phys. Lett. A 104 (3) pp 131– (1984)
[5] Au-Yang, Critical correlations in a Z-invariant inhomogeneous Ising model, Phys. A 144 (1) pp 44– (1987)
[6] Baik, On the distribution of the length of the longest increasingsubsequence of random permutations, J. Amer. Math. Soc. 12 (4) pp 1119– (1999) · Zbl 0932.05001
[7] Baik , J. Liu , Z. Discrete Toeplitz/Hankel determinants and the width of non-intersecting processes · Zbl 1320.60118
[8] Barnes, The theory of the G function, Quart. J. Pure and Appl. Math. 31 pp 264– (1900) · JFM 30.0389.02
[9] Basor, Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc. 239 pp 33– (1978) · Zbl 0409.47018
[10] Basor, A localization theorem for Toeplitz determinants, Indiana Univ. Math. J. 28 (6) pp 975– (1979) · Zbl 0396.47018
[11] Basor, Trace formulas for Toeplitz matrices with piecewise continuous symbols, J. Math. Anal. Appl. 120 (1) pp 25– (1986) · Zbl 0609.47036
[12] Basor, Toeplitz determinants from compatibility conditions, Ramanujan J. 16 (1) pp 25– (2008) · Zbl 1149.15003
[13] Basor, Asymptotics of block Toeplitz determinants and the classical dimer model, Comm. Math. Phys. 274 (2) pp 427– (2007) · Zbl 1128.47028
[14] Basor, A new proof of the Szego limit theorem and new results for Toeplitzoperators with discontinuous symbol, J. Operator Theory 3 (1) pp 23– (1980)
[15] Basor, The Fisher-Hartwig conjecture and generalizations. Current problems in statistical mechanics (Washington, 1991), Phys. A 177 (1-3) pp 167– (1991)
[16] Basor, On a Toeplitz determinant identity of Borodin and Okounkov, Integral Equations Operator Theory 37 (4) pp 397– (2000) · Zbl 0967.47020
[17] Baxter, A convergence equivalence related to polynomials orthogonal on the unit circle, Trans. Amer. Math. Soc. 99 pp 471– (1961) · Zbl 0116.35705
[18] Baxter, A norm inequality for a ”finite-section” Wiener-Hopf equation, Illinois J. Math. 7 pp 97– (1963) · Zbl 0113.09101
[19] Baxter, Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model, J. Stat. Phys. 145 (3) pp 518– (2011) · Zbl 1252.82024
[20] Baxter , R. J. Onsager and Kaufman’s calculations of the sponteaneous magnetization of the Ising model: II · Zbl 1257.82013
[21] Belokolos, Algebro-geometric approach to nonlinear evolution equations (1994) · Zbl 0809.35001
[22] Benettin, On the Onsager-Yang-value of the spontaneous magnetization, Comm. Math. Phys. 30 (1) pp 45– (1973)
[23] Borodin, A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory 37 (4) pp 386– (2000) · Zbl 0970.47014
[24] Böttcher , A. An operator theoretic approach to the brickwork Isingmodel with second-neighbor interactions. Personal communication 1992
[25] Böttcher, The Onsager formula, the Fisher-Hartwig conjecture, and their influence on research into Toeplitzoperators, J. Statist. Phys. 78 (1-2) pp 575– (1995) · Zbl 1080.47502
[26] Böttcher, Spectral properties of banded Toeplitz matrices (2005) · Zbl 1089.47001
[27] Böttcher, Inside the eigenvalues of certain Hermitian Toeplitz band matrices, J. Comput. Appl. Math. 233 (9) pp 2245– (2010) · Zbl 1195.15009
[28] Böttcher, The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral orders, itMath. Nachr. 102 (1981)
[29] Böttcher, Invertibility and asymptotics of Toeplitz matrices (1983) · Zbl 0578.47015
[30] Böttcher, Toeplitz matrices and determinants with Fisher-Hartwig symbols, J. Funct. Anal. 63 (2) pp 178– (1985) · Zbl 0592.47016
[31] Böttcher, Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity, Math. Nachr. 127 pp 95– (1986) · Zbl 0613.47024
[32] Böttcher, Introduction to large truncated Toeplitz matrices (1999)
[33] Böttcher, Analysis of Toeplitz operators (2006)
[34] Böttcher, Two elementary derivations of the pure Fisher-Hartwig determinant, Integral Equations Operator Theory 53 (4) pp 593– (2005) · Zbl 1081.47033
[35] Böttcher, Szego via Jacobi, Linear Algebra Appl. 419 (2-3) pp 656– (2006) · Zbl 1116.47024
[36] Bui, On the mean values of L-functions in orthogonal and symplectic families, Proc. Lond. Math. Soc. (3) 96 (2) pp 335– (2008) · Zbl 1220.11099
[37] Bump, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) pp 252– (2002) · Zbl 1005.47030
[38] Burgoyne, Remarks on the combinatorial approach to the Ising problem, J. Mathematical Phys. 4 (10) pp 1320– (1963) · Zbl 0151.46602
[39] Cafasso, Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies, Math. Phys. Anal. Geom. 11 (1) pp 11– (2007) · Zbl 1163.37022
[40] odory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, diegegebene Werte nicht annehmen, Math. Ann. 64 (1) pp 95– (1907) · JFM 38.0448.01
[41] odory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1) pp 193– (1911) · JFM 42.0429.01
[42] Carathéodory, Über den Zusammenhang der extremen von harmonischen Funktionen mit ihren Koeffizienten undüber den Picard-Landauschen Satz, Rend. Circ. Mat. Palermo 32 (1) pp 218– (1911) · JFM 42.0430.01
[43] Chang, The spontaneous magnetization of a two-dimensional rectangular Ising model, Phys. Rev. 88 (6) pp 1422– (1952)
[44] Chelkak , D. Hongler , C. Izyurov , K. Conformal invariance of spin correlations in the planar Ising model · Zbl 1318.82006
[45] Cheng, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. III, Phys. Rev. 164 (2) pp 719– (1967)
[46] Claeys, Emergence of a singularity for Toeplitz determinants and Painlevé V, Duke Math. J. 160 (2) pp 207– (2011) · Zbl 1298.47039
[47] Claeys , T. Krasovsky , I. Toeplitz determinants with merging singularities. In preparation · Zbl 1333.15018
[48] Clancey, Factorization of matrix functions and singular integral operators (1981) · Zbl 0474.47023
[49] Deift, Integrable operators. Differential operators and spectral theory pp 69– (1999)
[50] Deift, Orthogonal polynomials and random matrices: A Riemann-Hilbert approach (1999) · Zbl 0997.47033
[51] Deift , P. Universality for mathematical and physical systems International Congress of Mathematics 1 125 152 · Zbl 1149.82012
[52] Deift, Asymptotics of Toeplitz, Hankel and Toeplitz+Hankel determinants with Fisher-Hartwigsingularities, Ann. of Math. (2) 174 (2) pp 1243– (2011) · Zbl 1232.15006
[53] Deift, Eigenvalues of Toeplitz matrices in the bulk of thespectrum, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (4) pp 437– (2012)
[54] Deift , P. Its , A. Krasovsky , I. On the asymptotics of a Toeplitz determinant withsingularities. Preprint 2012 · Zbl 1326.35218
[55] Deift, The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math. 202 (1) pp 26– (2007) · Zbl 1116.15019
[56] Deift, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. (2) 146 (1) pp 149– (1997) · Zbl 0936.47028
[57] Deift, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (12) pp 1491– (1999) · Zbl 1026.42024
[58] Deift, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrixtheory, Comm. Pure Appl. Math. 52 (11) pp 1335– (1999) · Zbl 0944.42013
[59] Deift, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices 1997 (6) pp 286– (1997) · Zbl 0873.65111
[60] Deift, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MkdV equation, Ann. of Math. (2) 137 (2) pp 295– (1993) · Zbl 0771.35042
[61] Devinatz, The strong Szego limit theorem, Illinois J. Math. 11 pp 160– (1967) · Zbl 0166.40301
[62] Domb, On the theory of cooperative phenomena in crystals, Adv. in Phys. 9 (35) pp 245– (1960)
[63] Domb, The collected works of LarsOnsager pp 167– (1996)
[64] Douglas, Banach algebra techniques in operator theory (1998) · Zbl 0920.47001
[65] Dubédat , J. Exact bosonization of the Ising model · Zbl 1204.60079
[66] Duits, Powers of large random unitary matrices and Toeplitz determinants, Trans. Amer. Math. Soc. 362 (3) pp 1169– (2010) · Zbl 1187.60003
[67] Dym, Fourier series and integrals (1972)
[68] Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962) · Zbl 0105.41604
[69] Dyson, Statistical theory of the energy levels of complex systems. II, J. Mathematical Phys. 3 (1962) · Zbl 0105.41604
[70] Dyson , F. J. ”Toeplitz determinants and Coulomb gases.” Presentation to Eastern Theoretical Physics Conference Chapel Hill N.C.
[71] Dyson, Fredholm determinants and inverse scatteringproblems, Comm. Math. Phys. 47 (2) pp 171– (1976) · Zbl 0323.33008
[72] Dyson, Chen Ning Yang pp 131– (1995)
[73] Ehrhardt, Recent advances in operator theory (Groningen, 1998) pp 217– (2001)
[74] Faddeyev, The inverse problem in the quantum theory ofscattering. (Russian), Uspehi Mat. Nauk 14 (4) pp 57– (1959)
[75] Fisher, The susceptibility of the plane Ising model, Physica 25 (1-6) pp 521– (1959)
[76] Fisher, On the theory of critical point density fluctuations, Physica 28 (2) pp 172– (1962)
[77] Fisher, On the dimer solution of planar Ising models, J. Math. Phys. 7 (10) pp 1776– (1966)
[78] Fisher, Toeplitz determinants: some applications, theorems and conjectures, Adv. Chem. Phys. 15 pp 333– (2007)
[79] Fokas, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (2) pp 395– (1992) · Zbl 0760.35051
[80] Fokas, Painlevé transcendents (2006)
[81] Forrester, Applications and generalizations of Fisher-Hartwig asymptotics, J. Math. Phys. 45 (5) pp 2003– (2004) · Zbl 1071.82014
[82] Forrester, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (4) pp 489– (2008) · Zbl 1154.33002
[83] Franchini, Asymptotics of Toeplitz determinants and the emptiness formation probability for the XY spin chain, J. Phys. A: Math. Gen. 38 (23) pp 5069– (2005) · Zbl 1256.82004
[84] Gallavotti, Boundary conditions and correlation functions in the {\(\nu\)}-dimensional Ising model at low temperature, Comm. Math. Phys. 23 (4) pp 275– (1971) · Zbl 0236.60072
[85] Geronimo, Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys. 20 (2) pp 299– (1979) · Zbl 0439.33014
[86] Geronimus, Polynomials orthogonal on a circle and interval (1960)
[87] Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (2) pp 257– (1990) · Zbl 0704.05001
[88] Golinskiĭ, On Szegö’s limit theorem, Math. USSR Izv. 5 (2) pp 421– (1971) · Zbl 0249.42012
[89] Gonek, A hybrid Euler-Hadamard product for the Riemann zeta function, Duke Math. J. 136 (3) pp 507– (2007)
[90] Grenander, Toeplitz forms and their applications (1958) · Zbl 0080.09501
[91] Griffiths, Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet, Phys. Rev. (2) 136 (2A) pp A437– (1964) · Zbl 0129.23205
[92] Gutman, Non-equilibrium 1D many-body problems and asymptotic properties of Toeplitz determinants, J. Phys. A 44 (16) pp 21– (2011) · Zbl 1251.82032
[93] Gyires, Eigenwerte verallgemeinerter Toeplitzschen Matrizen, Publ. Math. Debrecen 4 pp 171– (1956)
[94] Hartwig, Asymptotic behavior of Toeplitz matrices and determinants, Arch. Rational Mech. Anal. 32 (3) pp 190– (1969) · Zbl 0169.04403
[95] Helton, Proceedings of a conference operator theory (Dalhousie University, Halifax, N.S., 1973) pp 141– (1973)
[96] Hirschman, On a theorem of Szego, Kac, and Baxter, J. Analyse Math. 14 (1) pp 225– (1965) · Zbl 0141.07001
[97] Hurst, New solution of the Ising problem for a rectangular lattice, J. Chem. Phys. 33 (4) pp 1059– (1960)
[98] Ibragimov, A theorem of Gabor Szegö, Mat. Zametki 3 pp 693– (1968)
[99] Ince, Ordinary differential equations (1956)
[100] Ising, Beitrag zur theorie des ferromagnetismus, Z. Phys. 31 (1) pp 253– (1925)
[101] Its, Universality and renormalization pp 151– (2007)
[102] Its, The Fisher-Hartwig formula and entanglemententropy, J. Stat. Phys. 137 (5-6) pp 1014– (2009) · Zbl 1183.82014
[103] Its, Entanglement entropy in quantum spin chains with finite range interaction, Comm. Math. Phys. 284 (1) pp 117– (2008) · Zbl 1157.82026
[104] Its, Random words, Toeplitz determinants and integrablesystems. II, Phys. D 152/153 pp 199– (2001) · Zbl 0977.35103
[105] Jimbo, Studies on holonomic quantum fields. XVII, Proc. Japan Acad. Ser. A Math. Sci. 56 (9) pp 405– (1980)
[106] Jimbo, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1) pp 80– (1980) · Zbl 1194.82007
[107] Jin, Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture, J. Statist. Phys. 116 (1-4) pp 79– (2004) · Zbl 1142.82314
[108] Johansson, On Szegö/s asymptotic formula for Toeplitz determinants and generalizations, Bull. Sci. Math. (2) 112 (3) pp 257– (1988)
[109] Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1) pp 151– (1998) · Zbl 1039.82504
[110] Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21 (3) pp 501– (1954) · Zbl 0056.10201
[111] Kac, A combinatorial solution of the two-dimensional Ising model, Phys. Rev. 88 (6) pp 1332– (1952) · Zbl 0048.45804
[112] Kadanoff, Spin-spin correlations in the two-dimensional Ising model, Il Nuovo Cimento B 44 (2) pp 276– (1966)
[113] Kasteleyn, The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica 27 (12) pp 1209– (1961) · Zbl 1244.82014
[114] Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 (2) pp 287– (1963)
[115] Kaufman, Crystal statistics II. Partition function evaluated by spinor analysis, Phys. Rev. 76 (8) pp 1232– (1949) · Zbl 0035.42801
[116] Kaufman, Letter from Lars Onsager to Bruria Kaufman, J. Statist. Phys. 78 (1995) · Zbl 1102.01311
[117] Kaufman, Crystal statistics. III. Short-range order in a binary Ising lattice, Phys. Rev. 76 (8) pp 1244– (1949) · Zbl 0035.42802
[118] Keating, Applications of random matrices in physics pp 1– (2006)
[119] Keating, Random matrix theory and entanglement in quantum spin chains, Comm. Math. Phys. 252 (1-3) pp 543– (2004) · Zbl 1124.82009
[120] Kramers, Statistics of the two-dimensional ferromagnet. I, Phys. Rev. (2) 60 (3) pp 252– (1941) · Zbl 0027.28505
[121] Krasovsky, Random walks, boundaries and spectra pp 305– (2011) · Zbl 1220.47038
[122] Landau, Course of theoretical physics. Vol. 5: Statistical physics (1968)
[123] Lebowitz, Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems, Comm. Math. Phys. 28 (4) pp 313– (1972)
[124] Lebowitz, On the Uniqueness of the Equilibrium State for Ising Spin Systems, Commun. Math. Phys. 25 (4) pp 276– (1972)
[125] Lee, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Rev. (2) 87 pp 410– (1952) · Zbl 0048.43401
[126] Lenard, Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons, J. Mathematical Phys. 5 pp 930– (1964)
[127] Lenard, Some remarks on large Toeplitz determinants, Pacific J. Math. 42 pp 137– (1972) · Zbl 0255.42005
[128] Levitin, Spectral pollution and second-order relative spectra for self-adjoint operators, IMA J. Numer. Anal. 24 (3) pp 393– (2004) · Zbl 1060.65056
[129] Levitin, Operator theory and its applications pp 115– (2010)
[130] Lieb, Some of the early history of exactly soluble models, Internat. J. Modern Phys. B 11 pp 1– (1997) · Zbl 1229.82073
[131] Lieb, Fluxes, Laplacians, and Kasteleyn’s theorem, Duke Math. J. 71 (2) pp 337– (1993) · Zbl 0787.05083
[132] Litvinchuk, Factorization of measurable matrix functions (1987)
[133] Lyberg, Form factor expansion of the row and diagonal correlation functions of the two-dimensional Ising model, J. Phys. A 40 (13) pp 3329– (2007) · Zbl 1114.82009
[134] McCoy, Gabor Szegö: Collected papers, Volume 1: 1915-1927 pp 47– (1982)
[135] McCoy , B. M. The romance of the Ising model. Preprint 2011 · Zbl 1270.81107
[136] McCoy, Relation of conformal field theory and deformation theory for the Ising model, Nuclear Phys. B 285 (2) pp 279– (1987)
[137] McCoy, Painlevé functions of thethird kind, J. Mathematical Phys. 18 (5) pp 1058– (1977) · Zbl 0353.33008
[138] McCoy, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. II, Phys. Rev. 155 (2) pp 438– (1967)
[139] McCoy, The two-dimensional Ising model (1973) · Zbl 1094.82500
[140] Mehta, Random matrices (2004)
[141] Montroll, Correlations and spontaneous magnetization of the two-dimensional Ising model, J. Mathematical Phys. 4 (2) pp 308– (1963)
[142] Myers, Wave scattering and the geometry of a strip, J. Mathematical Phys. 6 pp 1839– (1965)
[143] Nickel, On the singularity structure of the 2D Ising model susceptibility, J. Phys. A 32 (21) pp 3889– (1999) · Zbl 0936.82006
[144] Nickel, Addendum to: ”On the singularity structure of the 2D Ising model susceptibility.”, J. Phys. A 33 (8) pp 1693– (2000) · Zbl 0967.82006
[145] Niles , D. The Riemann-Hilbert -Birkhoff inverse monodromy problem and connectionformulae for the third Painlevé transcendent. Doctoral dissertation,Indiana University/Purdue University at Indianapolis 2009
[146] Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. (2) 65 pp 117– (1944) · Zbl 0060.46001
[147] Onsager, Critical phenomena in alloys, magnets, and superconductors, xix-xxiv (1971)
[148] Onsager, Critical phenomena in alloys, magnets, and superconductors pp 3– (1971)
[149] Ovchinnikov, Fisher-Hartwig conjecture and the correlators in the impenetrable Bose gas, Phys. Lett. A 373 (3) pp 305– (2009) · Zbl 1227.82070
[150] Palmer, Planar Ising correlations (2007) · Zbl 1136.82001
[151] Peierls, On Ising’s model for ferromagnetism, Proc. Camb. Phil. Soc. 32 (3) pp 477– (1936) · Zbl 0014.33604
[152] Peller, Hankel operators and their applications (2003)
[153] Penrose, Bose-Einstein condensation and liquidhelium, Phys. Rev. 104 pp 576– (1956) · Zbl 0071.44701
[154] Pinson , H. Rotational invariance of the 2d spin spin correlation function. Preprint 2011 · Zbl 1251.82022
[155] Potts, The combinatorial method and the two-dimensional Ising model, Progr. Theoret. Phys. 13 pp 38– (1955) · Zbl 0065.24001
[156] Ruelle, Statistical mechanics: Rigorous results (1969)
[157] Ryazanov, Correlation asymptotics for a plane Ising lattice, Sov. Phys. JETP 22 pp 789– (1966)
[158] Saff, Logarithmic potentials with external fields (1997) · Zbl 0881.31001
[159] Sakhnovich, Operators which are similar to unitary operators withabsolutely continuous spectrum, Functional Analysis and Its Applications 2 (1) pp 48– (1968)
[160] Schultz, Note on the one-dimensional gas of impenetrable point-particle bosons, J. Mathematical Phys. 4 pp 666– (1963)
[161] Schultz, Two-dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys. 36 (3) pp 856– (1964)
[162] Segur, Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent, Phys. D 3 pp 1– (1981) · Zbl 1194.35388
[163] Sherman, Combinatorial aspects of the Ising model for ferromagnetism. I. Aconjecture of Feynman on paths and graphs, J. Mathematical Phys. 1 (3) pp 202– (1960) · Zbl 0123.45501
[164] Shrock, Off-axis correlation functions in the isotropic d = 2 Ising model, Phys. Rev. B 31 (3) pp 1486– (1985)
[165] Simon, The statistical mechanics of lattice gases. Vol. I. (1993) · Zbl 0804.60093
[166] Simon, Orthogonal polynomials on the unit circle. Parts 1 and 2. (2005)
[167] Simon, Trace ideals and their applications (2005) · Zbl 1074.47001
[168] Szego, Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellenpositiven Funktion. (German), Math. Ann. 76 (4) pp 490– (1915) · JFM 45.0518.02
[169] Szego, Beiträge zur Theorie der Toeplitzschen Formen. (German), Math. Z. 6 pp 3– (1920)
[170] Szego, Beiträge zur Theorie der Toeplitzschen Formen, II, Math. Zeit. 9 pp 167– (1921)
[171] Szego, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 pp 228– (1952)
[172] Szego, Orthogonal polynomials (1975)
[173] Tanaka, Spin pair correlation function of the Ising model on the brickworklattice with second neighbor interactions, Phys. A 171 (2) pp 350– (1991)
[174] Temperley, Dimer problem in statistical mechanics-an exact result, Philos. Mag. (8) 6 pp 1061– (1961) · Zbl 0126.25102
[175] Toeplitz, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse pp 110– (1907)
[176] Toeplitz, Zur Theorie der quadratischen und bilinearen Formen vonunendlichvielen Veränderlichen. (German), Math. Ann. 70 (3) pp 351– (1911) · JFM 42.0366.01
[177] Toeplitz, Über die Fouriersche Entwickelung positiver Funktionen, Rend. Cire. Mat. Palermo 32 pp 191– (1911) · JFM 42.0428.02
[178] Tracy, Asymptotics of a {\(\tau\)}-function arising in the two-dimensional Ising model, Comm. Math. Phys. 142 (2) pp 297– (1991) · Zbl 0734.60106
[179] Tracy, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1) pp 151– (1994) · Zbl 0789.35152
[180] Vaidya, One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature, J. Mathematical Phys. 20 pp 2291– (1979)
[181] van der Waerden, Die lange Reichweite der regelmässigen Atomanordnung in Mischkristallen, Z. Physik 118 pp 7– (1941) · Zbl 0026.28301
[182] Vdovichenko, A calculation of the partition function for a plane dipole lattice, Ž. Éksper. Teoret. Fiz. 47 pp 715–
[183] Verblunsky, On positive harmonic functions, Proc. London Math. Soc. S2-40 (1) pp 290– (1936) · Zbl 0013.15702
[184] Vidal, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (22) pp 227902– (2003)
[185] Wegert, Constantin Carathéodory in his ... origins pp 61– (2001)
[186] Widom, Studies in real and complex analysis pp 179– (1965)
[187] Widom, The strong Szegö limit theorem for circular arcs, Indiana Univ. Math. J. 21 pp 277– · Zbl 0213.34903
[188] Widom, Toeplitz determinants with singular generating functions, Amer. J. Math. 95 (2) pp 333– (1973) · Zbl 0275.45006
[189] Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Advances in Math. 13 pp 284– (1974) · Zbl 0281.47018
[190] Widom, On the limit of block Toeplitz determinants, Proc. Amer. Math. Soc. 50 pp 167– (1975) · Zbl 0312.47027
[191] Widom , H. Asymptotic behavior of block Toeplitz matrices and determinants. II. Advances in Math 21 1976 1 1 29 · Zbl 0344.47016
[192] Widom, Topics in operator theory: Ernst D. Hellinger memorial volume pp 387– (1990)
[193] Widom, Integrable systems and random matrices pp 31– (2008)
[194] Witte, Isomonodromic deformation theory and the next-to-diagonal correlationsof the anisotropic square lattice Ising model, J. Phys. A 40 (24) pp F491– (2007) · Zbl 1116.82008
[195] Witte, Fredholm determinant evaluations of the Ising model diagonalcorrelations and their {\(\lambda\)}generalization, Stud. Appl. Math. 128 (2) pp 183– (2012) · Zbl 1250.82010
[196] Wu, Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model, I. Phys. Rev. 149 (1) pp 380– (1966)
[197] Wu, Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B 13 (1) pp 316– (1976)
[198] Yang, The spontaneous magnetization of a two-dimensional Ising model, Physical Rev. (2) 85 pp 808– (1952) · Zbl 0046.45304
[199] Yang, Selected Papers 1945-1980 with Commentary pp 11– (2005) · Zbl 1130.01019
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