Abstract
We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.
-
[belavin-polyakov-zamolodchikov-i]
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, "Infinite conformal symmetry in two-dimensional quantum field theory," Nuclear Phys. B, vol. 241, iss. 2, pp. 333-380, 1984.@article{belavin-polyakov-zamolodchikov-i, mrkey = {0757857},
author = {Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.},
title = {Infinite conformal symmetry in two-dimensional quantum field theory},
journal = {Nuclear Phys. B},
fjournal = {Nuclear Physics. B},
volume = {241},
year = {1984},
number = {2},
pages = {333--380},
issn = {0550-3213},
coden = {NUPBBO},
mrclass = {81E13 (17B70 58G37 81D15)},
mrnumber = {0757857},
doi = {10.1016/0550-3213(84)90052-X},
zblnumber = {0661.17013},
} -
[belavin-polyakov-zamolodchikov-ii] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, "Infinite conformal symmetry of critical fluctuations in two dimensions," J. Statist. Phys., vol. 34, iss. 5-6, pp. 763-774, 1984.
@article{belavin-polyakov-zamolodchikov-ii, mrkey = {0751712},
author = {Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.},
title = {Infinite conformal symmetry of critical fluctuations in two dimensions},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {34},
year = {1984},
number = {5-6},
pages = {763--774},
issn = {0022-4715},
coden = {JSTPSB},
mrclass = {82A25 (17B67 22E65 22E70 81E99 82A68 82A97)},
mrnumber = {0751712},
mrreviewer = {Alan D. Sokal},
doi = {10.1007/BF01009438},
} -
[boutillier-de-tiliere-ii] C. Boutillier and B. de Tilière, "The critical ${Z}$-invariant Ising model via dimers: the periodic case," Probab. Theory Related Fields, vol. 147, iss. 3-4, pp. 379-413, 2010.
@article{boutillier-de-tiliere-ii, mrkey = {2639710},
author = {Boutillier, C{é}dric and de Tili{è}re, B{é}atrice},
title = {The critical {${Z}$}-invariant {I}sing model via dimers: the periodic case},
journal = {Probab. Theory Related Fields},
fjournal = {Probability Theory and Related Fields},
volume = {147},
year = {2010},
number = {3-4},
pages = {379--413},
issn = {0178-8051},
coden = {PTRFEU},
mrclass = {82B20 (05C90)},
mrnumber = {2639710},
mrreviewer = {Martin V. Hildebrand},
doi = {10.1007/s00440-009-0210-1},
zblnumber = {1195.82011},
} -
[boutillier-de-tiliere-i] C. Boutillier and B. de Tilière, "The critical $Z$-invariant Ising model via dimers: locality property," Comm. Math. Phys., vol. 301, iss. 2, pp. 473-516, 2011.
@article{boutillier-de-tiliere-i, mrkey = {2764995},
author = {Boutillier, C{é}dric and de Tili{è}re, B{é}atrice},
title = {The critical {$Z$}-invariant {I}sing model via dimers: locality property},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {301},
year = {2011},
number = {2},
pages = {473--516},
issn = {0010-3616},
coden = {CMPHAY},
mrclass = {82B20 (31C20 60K35)},
mrnumber = {2764995},
mrreviewer = {Eugenii Shustin},
doi = {10.1007/s00220-010-1151-3},
zblnumber = {1245.05027},
} -
[burkhard-guim-i] T. W. Burkhardt and I. Guim, "Bulk, surface, and interface properties of the Ising model and conformal invariance," Phys. Rev. B, vol. 36, iss. 4, pp. 2080-2083, 1987.
@article{burkhard-guim-i, mrkey = {0903407},
author = {Burkhardt, Theodore W. and Guim, Ihnsouk},
title = {Bulk, surface, and interface properties of the {I}sing model and conformal invariance},
journal = {Phys. Rev. B},
fjournal = {Physical Review. B. Condensed Matter. Third Series},
volume = {36},
year = {1987},
number = {4},
pages = {2080--2083},
issn = {0163-1829},
coden = {PRBMDO},
mrclass = {82A68 (82A25)},
mrnumber = {0903407},
doi = {10.1103/PhysRevB.36.2080},
} -
[burkhard-guim] T. W. Burkhardt and I. Guim, "Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordered boundary fields," Phys. Rev. B, vol. 47, pp. 14306-14311, 1993.
@article{burkhard-guim,
author = {Burkhardt, Theodore W. and Guim, Ihnsouk},
title = {Conformal theory of the two-dimensional {I}sing model with homogeneous boundary conditions and with disordered boundary fields},
journal = {Phys. Rev. B},
volume = {47},
pages = {14306--14311},
year = {1993},
doi = {10.1103/PhysRevB.47.14306},
} -
[camia-garban-newman-i] F. Camia, C. Garban, and C. M. Newman, "Planar Ising magnetization field I. Uniqueness of the critical scaling limit," Ann. Probab., vol. 43, iss. 2, pp. 528-571, 2015.
@article{camia-garban-newman-i, mrkey={3305999},
AUTHOR = {Camia, Federico and Garban, Christophe and Newman, Charles M.},
TITLE = {Planar {I}sing magnetization field {I}. {U}niqueness of the critical scaling limit},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {43},
YEAR = {2015},
NUMBER = {2},
PAGES = {528--571},
ISSN = {0091-1798},
MRCLASS = {82B20 60G20 60G60 60K35 82B27)},
MRNUMBER = {3305999},
DOI = {10.1214/13-AOP881},
} -
[cardy-i] J. L. Cardy, "Conformal invariance and surface critical behavior," Nucl. Phys. B, vol. 240, pp. 514-532, 1984.
@article{cardy-i,
author = {Cardy, J. L.},
title = {Conformal invariance and surface critical behavior},
journal = {Nucl. Phys. B},
volume = {240},
pages = {514--532},
year = {1984},
doi = {10.1016/0550-3213(84)90241-4},
} -
[chelkak-hongler] D. Chelkak and C. Hongler.
@misc{chelkak-hongler,
author = {Chelkak, Dmitry and Hongler, C.},
note = {in preparation},
} -
[chelkak-izyurov] D. Chelkak and K. Izyurov, "Holomorphic spinor observables in the critical Ising model," Comm. Math. Phys., vol. 322, iss. 2, pp. 303-332, 2013.
@article{chelkak-izyurov, mrkey = {3077917},
author = {Chelkak, Dmitry and Izyurov, Konstantin},
title = {Holomorphic spinor observables in the critical {I}sing model},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {322},
year = {2013},
number = {2},
pages = {303--332},
issn = {0010-3616},
mrclass = {82B20 (81R25 82B27)},
mrnumber = {3077917},
doi = {10.1007/s00220-013-1763-5},
zblnumber = {1277.82010},
} -
[chelkak-smirnov-i] D. Chelkak and S. Smirnov, "Discrete complex analysis on isoradial graphs," Adv. Math., vol. 228, iss. 3, pp. 1590-1630, 2011.
@article{chelkak-smirnov-i, mrkey = {2824564},
author = {Chelkak, Dmitry and Smirnov, Stanislav},
title = {Discrete complex analysis on isoradial graphs},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {228},
year = {2011},
number = {3},
pages = {1590--1630},
issn = {0001-8708},
coden = {ADMTA4},
mrclass = {60G50 (31C20 39A45 52C20 60J67)},
mrnumber = {2824564},
mrreviewer = {Ben Dyhr},
doi = {10.1016/j.aim.2011.06.025},
zblnumber = {1227.31011},
} -
[CDHKS] D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen, and S. Smirnov, "Convergence of Ising interfaces to Schramm’s SLE curves," C. R. Acad. Sci. Paris, Ser. I, vol. 352, pp. 157-161, 2014.
@article{CDHKS,
author = {Chelkak, D. and Duminil-Copin, H. and Hongler, C. and Kemppainen, A. and Smirnov, S.},
title = {Convergence of {I}sing interfaces to {S}chramm's {SLE }curves},
journal = {C. R. Acad. Sci. Paris, Ser. I},
volume = {352},
pages = {157--161},
year = {2014},
mrnumber = {3151886},
zblnumber = {1294.82007},
} -
[chelkak-smirnov-ii] D. Chelkak and S. Smirnov, "Universality in the 2D Ising model and conformal invariance of fermionic observables," Invent. Math., vol. 189, iss. 3, pp. 515-580, 2012.
@article{chelkak-smirnov-ii, mrkey = {2957303},
author = {Chelkak, Dmitry and Smirnov, Stanislav},
title = {Universality in the 2{D} {I}sing model and conformal invariance of fermionic observables},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {189},
year = {2012},
number = {3},
pages = {515--580},
issn = {0020-9910},
coden = {INVMBH},
mrclass = {82B20 (30C20 30C35 60J67 60K35)},
mrnumber = {2957303},
mrreviewer = {Ben Dyhr},
doi = {10.1007/s00222-011-0371-2},
zblnumber = {1257.82020},
} -
[dembo-montanari] A. Dembo and A. Montanari, "Ising models on locally tree-like graphs," Ann. Appl. Probab., vol. 20, iss. 2, pp. 565-592, 2010.
@article{dembo-montanari, mrkey = {2650042},
author = {Dembo, Amir and Montanari, Andrea},
title = {Ising models on locally tree-like graphs},
journal = {Ann. Appl. Probab.},
fjournal = {The Annals of Applied Probability},
volume = {20},
year = {2010},
number = {2},
pages = {565--592},
issn = {1050-5164},
mrclass = {82B44 (05C05 05C80 60F10 60K35 82B23)},
mrnumber = {2650042},
mrreviewer = {David J. Aldous},
doi = {10.1214/09-AAP627},
zblnumber = {1191.82025},
} -
[difrancesco-saleur-zuber] P. Di Francesco, H. Saleur, and J. -B. Zuber, "Critical Ising correlation functions in the plane and on the torus," Nuclear Phys. B, vol. 290, iss. 4, pp. 527-581, 1987.
@article{difrancesco-saleur-zuber, mrkey = {0922845},
author = {Di Francesco, P. and Saleur, H. and Zuber, J.-B.},
title = {Critical {I}sing correlation functions in the plane and on the torus},
journal = {Nuclear Phys. B},
fjournal = {Nuclear Physics. B},
volume = {290},
year = {1987},
number = {4},
pages = {527--581},
issn = {0550-3213},
coden = {NUPBBO},
mrclass = {82A25 (81E99 82A68)},
mrnumber = {0922845},
mrreviewer = {M. Lawrence Glasser},
doi = {10.1016/0550-3213(87)90202-1},
} -
[dubedat-i] J. Dubédat, Dimers and analytic torsion I, 2011.
@misc{dubedat-i,
author = {Dubédat, J.},
title = {Dimers and analytic torsion {I}},
arxiv = {1110.2808},
year = {2011},
} -
[dubedat-ii] J. Dubédat, Exact bosonization of the Ising model, 2011.
@misc{dubedat-ii,
author = {Dubédat, J.},
title = {Exact bosonization of the {I}sing model},
arxiv = {1112.4399},
year = {2011},
} -
[duminil-copin-hongler-nolin] H. Duminil-Copin, C. Hongler, and P. Nolin, "Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model," Comm. Pure Appl. Math., vol. 64, iss. 9, pp. 1165-1198, 2011.
@article{duminil-copin-hongler-nolin, mrkey = {2839298},
author = {Duminil-Copin, Hugo and Hongler, Cl{é}ment and Nolin, Pierre},
title = {Connection probabilities and {RSW}-type bounds for the two-dimensional {FK} {I}sing model},
journal = {Comm. Pure Appl. Math.},
fjournal = {Communications on Pure and Applied Mathematics},
volume = {64},
year = {2011},
number = {9},
pages = {1165--1198},
issn = {0010-3640},
coden = {CPAMA},
mrclass = {60K35 (82B20 82B43)},
mrnumber = {2839298},
mrreviewer = {Andrew R. Wade},
doi = {10.1002/cpa.20370},
zblnumber = {1227.82015},
} -
[gheissari-hongler-park] R. Gheissari, C. Hongler, and S. Park, Conformal Invariance of Spin Pattern Probabilities in the Planar Ising Model, 2013.
@misc{gheissari-hongler-park,
author = {Gheissari, R. and Hongler, C. and Park, S.},
title = {Conformal Invariance of Spin Pattern Probabilities in the Planar {I}sing Model},
arxiv = {1312.4446},
year = {2013},
} -
[griffith-hurst-sherman] R. B. Griffiths, C. A. Hurst, and S. Sherman, "Concavity of magnetization of an Ising ferromagnet in a positive external field," J. Mathematical Phys., vol. 11, pp. 790-795, 1970.
@article{griffith-hurst-sherman, mrkey = {0266507},
author = {Griffiths, Robert B. and Hurst, C. A. and Sherman, S.},
title = {Concavity of magnetization of an {I}sing ferromagnet in a positive external field},
journal = {J. Mathematical Phys.},
fjournal = {Journal of Mathematical Physics},
volume = {11},
year = {1970},
pages = {790--795},
issn = {0022-2488},
mrclass = {81.05},
mrnumber = {0266507},
doi = {10.1063/1.1665211},
} -
[grimmett] G. Grimmett, The Random-Cluster Model, New York: Springer-Verlag, 2006, vol. 333.
@book{grimmett, mrkey = {2243761},
author = {Grimmett, Geoffrey},
title = {The Random-Cluster Model},
series = {Grundl. Math. Wissen.},
volume = {333},
publisher = {Springer-Verlag},
year = {2006},
pages = {xiv+377},
isbn = {978-3-540-32890-2; 3-540-32890-4},
mrclass = {60K35 (60-02 82-02 82B20 82B43)},
mrnumber = {2243761},
mrreviewer = {Olivier Garet},
doi = {10.1007/978-3-540-32891-9},
address = {New York},
zblnumber = {1122.60087},
} -
[hongler-i] C. Hongler, Conformal invariance of Ising model correlations, 2010.
@misc{hongler-i,
author = {Hongler, Cl{é}ment},
title = {Conformal invariance of {I}sing model correlations},
note = {Ph.D. thesis, University of Geneva},
url = {http://www.math.columbia.edu/~hongler/thesis.pdf},
year = {2010},
} -
[hongler-smirnov-ii] C. Hongler and S. Smirnov, "The energy density in the planar Ising model," Acta Math., vol. 211, iss. 2, pp. 191-225, 2013.
@article{hongler-smirnov-ii, mrkey = {3143889},
author = {Hongler, Cl{é}ment and Smirnov, Stanislav},
title = {The energy density in the planar {I}sing model},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {211},
year = {2013},
number = {2},
pages = {191--225},
issn = {0001-5962},
mrclass = {82B20},
mrnumber = {3143889},
mrreviewer = {Xifeng Su},
doi = {10.1007/s11511-013-0102-1},
zblnumber = {1287.82007},
} -
[hongler-icmp-2012] C. Hongler, "Conformal invariance of Ising model correlations," in XVIIth International Congress on Mathematical Physics, Hackensack, NJ: World Scientific Publ., 2014, pp. 326-335.
@incollection{hongler-icmp-2012, mrkey = {3204482},
author = {Hongler, Cl{é}ment},
title = {Conformal invariance of {I}sing model correlations},
booktitle = {\hbox{XVII}th {I}nternational {C}ongress on {M}athematical {P}hysics},
publisher = {World Scientific Publ.},
address = {Hackensack, NJ},
year = {2014},
mrclass = {82B20 (81T40)},
mrnumber = {3204482},
doi = {10.1142/9789814449243_0023},
pages = {326-335},
} -
[izyurov-freebc] K. Izyurov, Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities, 2014.
@misc{izyurov-freebc,
author = {Izyurov, K.},
title = {Smirnov's observable for free boundary conditions, interfaces and crossing probabilities},
arxiv = {1404.0239},
year = {2014},
} -
[kadanoff-ceva] L. P. Kadanoff and H. Ceva, "Determination of an operator algebra for the two-dimensional Ising model," Phys. Rev. B, vol. 3, pp. 3918-3939, 1971.
@article{kadanoff-ceva, mrkey = {0389111},
author = {Kadanoff, Leo P. and Ceva, Horacio},
title = {Determination of an operator algebra for the two-dimensional {I}sing model},
journal = {Phys. Rev. B},
fjournal = {Physical Review. B. Solid State},
volume = {3},
year = {1971},
pages = {3918--3939},
mrclass = {82.62},
mrnumber = {0389111},
mrreviewer = {P. Caldirola},
doi = {10.1103/PhysRevB.3.3918},
} -
[kesten] H. Kesten, "Hitting probabilities of random walks on ${\bf Z}^d$," Stochastic Process. Appl., vol. 25, iss. 2, pp. 165-184, 1987.
@article{kesten, mrkey = {0915132},
author = {Kesten, Harry},
title = {Hitting probabilities of random walks on {${\bf Z}\sp d$}},
journal = {Stochastic Process. Appl.},
fjournal = {Stochastic Processes and their Applications},
volume = {25},
year = {1987},
number = {2},
pages = {165--184},
issn = {0304-4149},
coden = {STOPB7},
mrclass = {60J15 (31C20 82A41 82A67)},
mrnumber = {0915132},
mrreviewer = {Gregory F. Lawler},
doi = {10.1016/0304-4149(87)90196-7},
zblnumber = {0626.60067},
} -
[kramers-wannier] H. A. Kramers and G. H. Wannier, "Statistics of the two-dimensional ferromagnet. Part I," Phys. Rev., vol. 60, pp. 252-262, 1941.
@article{kramers-wannier, mrkey = {0004803},
author = {Kramers, H. A. and Wannier, G. H.},
title = {Statistics of the two-dimensional ferromagnet. Part I},
journal = {Phys. Rev.},
volume = {60},
year = {1941},
pages = {252--262},
mrclass = {79.0X},
mrnumber = {0004803},
mrreviewer = {L. W. Nordheim},
zblnumber = {0027.28505},
doi = {10.1103/PhysRev.60.252},
} -
[lawler-limic] G. F. Lawler and V. Limic, "The Beurling estimate for a class of random walks," Electron. J. Probab., vol. 9, p. no. 27, 846-861, 2004.
@article{lawler-limic, mrkey = {2110020},
author = {Lawler, Gregory F. and Limic, Vlada},
title = {The {B}eurling estimate for a class of random walks},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {9},
year = {2004},
pages = {no. 27, 846--861},
issn = {1083-6489},
mrclass = {60G50 (60F99)},
mrnumber = {2110020},
mrreviewer = {Marc Peign{é}},
doi = {10.1214/EJP.v9-228},
zblnumber = {1063.60066},
} -
[mccoy-wu-i] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Cambridge, MA: Harvard Univ. Press, 1973.
@book{mccoy-wu-i,
author = {McCoy, B. M. and Wu, T. T.},
title = {The Two-Dimensional Ising Model},
publisher = {Harvard Univ. Press},
address = {Cambridge, MA},
year = {1973},
zblnumber = {1094.82500},
} -
[onsager] L. Onsager, "Crystal statistics. I. A two-dimensional model with an order-disorder transition," Phys. Rev., vol. 65, pp. 117-149, 1944.
@article{onsager, mrkey = {0010315},
author = {Onsager, Lars},
title = {Crystal statistics. {I}. {A} two-dimensional model with an order-disorder transition},
journal = {Phys. Rev.},
volume = {65},
year = {1944},
pages = {117--149},
mrclass = {82.1X},
mrnumber = {0010315},
mrreviewer = {J. S. Frame},
zblnumber = {0060.46001},
doi = {10.1103/PhysRev.65.117},
} -
[palmer] J. Palmer, Planar Ising Correlations, Boston: Birkhäuser, 2007, vol. 49.
@book{palmer, mrkey = {2332010},
author = {Palmer, John},
title = {Planar {I}sing Correlations},
series = {Progr. Math. Phys.},
volume = {49},
publisher = {Birkhäuser},
address = {Boston},
year = {2007},
pages = {xx+363},
isbn = {978-0-8176-4248-8},
mrclass = {82-02 (34M55 37K10 60K35 82B20 82B44)},
mrnumber = {2332010},
mrreviewer = {Jes{ú}s Salas},
zblnumber = {1136.82001},
} -
[palmer-tracy] J. Palmer and C. Tracy, "Two-dimensional Ising correlations: the SMJ analysis," Adv. in Appl. Math., vol. 4, iss. 1, pp. 46-102, 1983.
@article{palmer-tracy, mrkey = {0695532},
author = {Palmer, John and Tracy, Craig},
title = {Two-dimensional {I}sing correlations: the {SMJ} analysis},
journal = {Adv. in Appl. Math.},
fjournal = {Advances in Applied Mathematics},
volume = {4},
year = {1983},
number = {1},
pages = {46--102},
issn = {0196-8858},
mrclass = {82A05 (58A15 81E05 82A68)},
mrnumber = {0695532},
mrreviewer = {V. Rivasseau},
doi = {10.1016/0196-8858(83)90005-2},
zblnumber = {0542.46044},
} -
[pinson] H. Pinson, "Rotational invariance of the 2d spin-spin correlation function," Comm. Math. Phys., vol. 314, iss. 3, pp. 807-816, 2012.
@article{pinson, mrkey = {2964775},
author = {Pinson, Haru},
title = {Rotational invariance of the 2d spin-spin correlation function},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {314},
year = {2012},
number = {3},
pages = {807--816},
issn = {0010-3616},
coden = {CMPHAY},
mrclass = {82B20 (82B26)},
mrnumber = {2964775},
mrreviewer = {N. N. Ganikhodjaev},
doi = {10.1007/s00220-012-1545-5},
zblnumber = {1251.82022},
} -
[sato-miwa-jimbo-i] M. Sato, T. Miwa, and M. Jimbo, "Studies on holonomic quantum fields. I," Proc. Japan Acad. Ser. A Math. Sci., vol. 53, iss. 1, pp. 6-10, 1977.
@article{sato-miwa-jimbo-i, mrkey = {0674350},
author = {Sato, Mikio and Miwa, Tetsuji and Jimbo, Michio},
title = {Studies on holonomic quantum fields. {I}},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
fjournal = {Japan Academy. Proceedings. Series A. Mathematical Sciences},
volume = {53},
year = {1977},
number = {1},
pages = {6--10},
issn = {0386-2194},
mrclass = {81.58 (15A66 30A88 35Q15 58A15 58G15 82.58)},
mrnumber = {0674350},
mrreviewer = {P. D. F. Ion},
doi = {10.3792/pjaa.53.6},
} -
[sato-miwa-jimbo-ii] M. Sato, T. Miwa, and M. Jimbo, "Holonomic quantum fields. III," Publ. Res. Inst. Math. Sci., vol. 15, iss. 2, pp. 577-629, 1979.
@article{sato-miwa-jimbo-ii, mrkey = {0555666},
author = {Sato, Mikio and Miwa, Tetsuji and Jimbo, Michio},
title = {Holonomic quantum fields. {III}},
journal = {Publ. Res. Inst. Math. Sci.},
fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
volume = {15},
year = {1979},
number = {2},
pages = {577--629},
issn = {0034-5318},
coden = {KRMPBV},
mrclass = {81E08 (32D99 58G07 58G37 81C05)},
mrnumber = {0555666},
doi = {10.2977/prims/1195188185},
zblnumber = {0436.35076},
} -
[sato-miwa-jimbo-iii] M. Sato, T. Miwa, and M. Jimbo, "Holonomic quantum fields. IV," Publ. Res. Inst. Math. Sci., vol. 15, iss. 3, pp. 871-972, 1979.
@article{sato-miwa-jimbo-iii, mrkey = {0566086},
author = {Sato, Mikio and Miwa, Tetsuji and Jimbo, Michio},
title = {Holonomic quantum fields. {IV}},
journal = {Publ. Res. Inst. Math. Sci.},
fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
volume = {15},
year = {1979},
number = {3},
pages = {871--972},
issn = {0034-5318},
coden = {KRMPBV},
mrclass = {81E08 (58G37 81C05)},
mrnumber = {0566086},
mrreviewer = {Autorreferat},
doi = {10.2977/prims/1195187881},
zblnumber = {0436.35077},
} -
[sato-miwa-jimbo-iv] M. Sato, T. Miwa, and M. Jimbo, "Holonomic quantum fields. V," Publ. Res. Inst. Math. Sci., vol. 16, iss. 2, pp. 531-584, 1980.
@article{sato-miwa-jimbo-iv, mrkey = {0594916},
author = {Sato, Mikio and Miwa, Tetsuji and Jimbo, Michio},
title = {Holonomic quantum fields. {V}},
journal = {Publ. Res. Inst. Math. Sci.},
fjournal = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
volume = {16},
year = {1980},
number = {2},
pages = {531--584},
issn = {0034-5318},
coden = {KRMPBV},
mrclass = {81E08 (82A05 82A68)},
mrnumber = {0594916},
mrreviewer = {S. Vovk},
doi = {10.2977/prims/1195187216},
zblnumber = {0479.35072},
} -
[smirnov-i] S. Smirnov, "Towards conformal invariance of 2D lattice models," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421-1451.
@incollection{smirnov-i, mrkey = {2275653},
author = {Smirnov, Stanislav},
title = {Towards conformal invariance of 2{D} lattice models},
booktitle = {International {C}ongress of {M}athematicians. {V}ol. {II}},
pages = {1421--1451},
publisher = {Eur. Math. Soc., Zürich},
year = {2006},
mrclass = {82B20 (30C35 60K35 81T40 82B43)},
mrnumber = {2275653},
mrreviewer = {Roland M. Friedrich},
zblnumber = {1112.82014},
doi = {10.4171/022-2/68},
} -
[smirnov-ii] S. Smirnov, "Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model," Ann. of Math., vol. 172, iss. 2, pp. 1435-1467, 2010.
@article{smirnov-ii, mrkey = {2680496},
author = {Smirnov, Stanislav},
title = {Conformal invariance in random cluster models. {I}. {H}olomorphic fermions in the {I}sing model},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {172},
year = {2010},
number = {2},
pages = {1435--1467},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {60K35 (30G25 60J67 81T40 82B20)},
mrnumber = {2680496},
mrreviewer = {Roland M. Friedrich},
doi = {10.4007/annals.2010.172.1441},
zblnumber = {1200.82011},
} -
[yang] C. N. Yang, "The spontaneous magnetization of a two-dimensional Ising model," Physical Rev., vol. 85, pp. 808-816, 1952.
@article{yang, mrkey = {0051740},
author = {Yang, C. N.},
title = {The spontaneous magnetization of a two-dimensional {I}sing model},
journal = {Physical Rev.},
volume = {85},
year = {1952},
pages = {808--816},
mrclass = {80.1X},
mrnumber = {0051740},
mrreviewer = {F. J. Murray},
zblnumber = {0046.45304},
doi = {10.1103/PhysRev.85.808},
} -
[wu-mccoy-tracy-barouch] T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, "Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region," Phys. Rev. B, vol. 13, pp. 316-374, 1976.
@article{wu-mccoy-tracy-barouch,
author = {Wu, T. T. and McCoy, B. M. and Tracy, C. A. and Barouch, E.},
title = {Spin-spin correlation functions for the two-dimensional {I}sing model: Exact theory in the scaling region},
journal = {Phys. Rev. B},
volume = {13},
pages = {316--374},
year = {1976},
doi = {10.1103/PhysRevB.13.316},
}
Full Article