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Planar Ising magnetization field. I: Uniqueness of the critical scaling limit. (English) Zbl 1332.82012
This manuscript is devoted to the study of the renormalized magnetization field for the critical Ising model. The authors prove that the renormalized magnetization field, seen as a random distribution on the plane, has a scaling limit as the mesh size goes to \(0\) and that the limiting field is conformally covariant. They analyze the tightness of the magnetization field and then they present two proofs of the scaling limit. The first and second moments for the magnetization are studied in the appendix.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G20 Generalized stochastic processes
60G60 Random fields
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References:
[1] Abraham, D. B. (1973). Susceptibility and fluctuations in the Ising ferromagnet. Phys. Lett. A 43 163-164.
[2] Abraham, D. B. (1978). Pair function for the rectangular Ising ferromagnet. Comm. Math. Phys. 60 181-191.
[3] Abraham, D. B. and Reed, P. (1976). Interface profile of the Ising ferromagnet in two dimensions. Comm. Math. Phys. 49 35-46.
[4] Aizenman, M. (1996). The geometry of critical percolation and conformal invariance. In STATPHYS 19 ( Xiamen , 1995) 104-120. World Sci. Publ., River Edge, NJ.
[5] Aizenman, M. (1998). Scaling limit for the incipient spanning clusters. In Mathematics of Multiscale Materials ( Minneapolis , MN , 1995 - 1996) (K. Golden, G. Grimmett, R. James, G. Milton and P. Sen, eds.). IMA Vol. Math. Appl. 99 1-24. Springer, New York. · Zbl 0941.74013
[6] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419-453. · Zbl 0944.60022
[7] Belavin, A. A., Polyakov, A. M. and Zamolodchikov, A. B. (1984). Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34 763-774.
[8] Belavin, A. A., Polyakov, A. M. and Zamolodchikov, A. B. (1984). Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 333-380. · Zbl 0661.17013
[9] Camia, F. (2012). Towards conformal invariance and a geometric representation of the 2D Ising magnetization field. Markov Process. Related Fields 18 89-110. · Zbl 1260.82033
[10] Camia, F., Garban, C. and Newman, C. M. Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits. Preprint. Available at . arXiv:1307.3926 · Zbl 1338.82009
[11] Camia, F., Garban, C. and Newman, C. M. (2012). The Ising magnetization exponent is 1/15. Probab. Theory Related Fields . To appear. Available at . arXiv:1205.6612 · Zbl 1307.82004
[12] Camia, F. and Newman, C. M. (2004). Continuum nonsimple loops and 2D critical percolation. J. Stat. Phys. 116 157-173. · Zbl 1142.82332
[13] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1-38. · Zbl 1117.60086
[14] Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and \(\mathrm{SLE}_{6}\): A proof of convergence. Probab. Theory Related Fields 139 473-519. · Zbl 1126.82007
[15] Camia, F. and Newman, C. M. (2009). Ising (conformal) fields and cluster area measures. Proc. Natl. Acad. Sci. USA 106 5547-5463. · Zbl 1202.82017
[16] Chelkak, D., Duminil-Copin, H. and Hongler, C. Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Preprint. Available at . arXiv:1312.7785 · Zbl 1341.60124
[17] Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A. and Smirnov, S. (2014). Convergence of Ising interfaces to Schramm’s SLE curves. C.R. Acad. Sci. Paris Sér. I Math. 352 157-161. · Zbl 1294.82007
[18] Chelkak, D., Hongler, C. and Izyurov, K. (2012). Conformal invariance of spin correlations in the planar Ising model. Preprint. Available at . arXiv:1202.2838 · Zbl 1318.82006
[19] De Coninck, J. (1987). On limit theorems for the bivariate (magnetization, energy) variable at the critical point. Comm. Math. Phys. 109 191-205. · Zbl 0619.60096
[20] De Coninck, J. and Newman, C. M. (1990). The magnetization-energy scaling limit in high dimension. J. Stat. Phys. 59 1451-1467. · Zbl 0718.60116
[21] Di Francesco, P., Mathieu, P. and Sénéchal, D. (1997). Conformal Field Theory . Springer, New York. · Zbl 0869.53052
[22] Dubédat, J. (2009). SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 995-1054. · Zbl 1204.60079
[23] Dubédat, J. (2011). Exact bosonization of the Ising model. Preprint. Available at . arXiv:1112.4399
[24] Duminil-Copin, H., Hongler, C. and Nolin, P. (2011). Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 1165-1198. · Zbl 1227.82015
[25] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536-564.
[26] Garban, C., Pete, G. and Schramm, O. The scaling limits of near-critical and dynamical percolation. Preprint. Available at . arXiv:1305.5526 · Zbl 1276.60111
[27] Garban, C., Pete, G. and Schramm, O. (2013). Pivotal, cluster, and interface measures for critical planar percolation. J. Amer. Math. Soc. 26 939-1024. · Zbl 1276.60111
[28] Griffiths, R. B., Hurst, C. A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 790-795.
[29] Grimmett, G. (2006). The Random-cluster Model. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 333 . Springer, Berlin. · Zbl 0858.60093
[30] Hongler, C. (2010). Conformal invariance of Ising model correlations. Ph.D. dissertation, Univ. Geneva. · Zbl 1304.82013
[31] Hongler, C. and Smirnov, S. (2013). The energy density in the planar Ising model. Acta Math. 211 191-225. · Zbl 1287.82007
[32] Kadanoff, L. P. and Ceva, H. (1971). Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B 3 3918-3939.
[33] Kasteleyn, P. W. and Fortuin, C. M. (1969). Phase transitions in lattice systems with random local properties. J. Phys. Soc. Jpn. , Suppl. 26 11-14.
[34] Kemppainen, A. (2009). On random planar curves and their scaling limits. Ph.D. dissertation, Univ. Helsinki.
[35] Kemppainen, A. and Smirnov, S. Random curves, scaling limits and Loewner evolutions. Preprint. Available at . arXiv:1212.6215 · Zbl 1393.60016
[36] Kemppainen, A. and Smirnov, S. Conformal invariance in random cluster models. II. Full scaling limit. Unpublished manuscript.
[37] Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759-795. · Zbl 1043.52014
[38] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128-1137. · Zbl 1034.82021
[39] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011
[40] Ledoux, M. and Talagrand, M. (2011). Probability in Banach Spaces . Springer, Berlin. Isoperimetry and processes, Reprint of the 1991 edition. · Zbl 1226.60003
[41] McCoy, B. M. and Wu, T. T. (1973). The Two-Dimensional Ising Model . Harvard Univ. Press, Cambridge, MA. · Zbl 1094.82500
[42] Mercat, C. (2001). Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218 177-216. · Zbl 1043.82005
[43] Newman, C. M. (1975). Gaussian correlation inequalities for ferromagnets. Z. Wahrsch. Verw. Gebiete 33 75-93. · Zbl 0297.60053
[44] Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65 117-149. · Zbl 0060.46001
[45] Palmer, J. (2007). Planar Ising Correlations . Birkhäuser, Boston, MA. · Zbl 1136.82001
[46] Polyakov, A. M. (1970). Conformal symmetry of critical fluctuations. JETP Letters 12 381-383.
[47] Riva, V. and Cardy, J. (2006). Holomorphic parafermions in the Potts model and stochastic Loewner evolution. J. Stat. Mech. Theory Exp. 12 P12001, 19 pp. (electronic).
[48] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093
[49] Schramm, O. and Smirnov, S. (2011). On the scaling limits of planar percolation. Ann. Probab. 39 1768-1814. · Zbl 1231.60116
[50] Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79-129. · Zbl 1170.60008
[51] Sheffield, S. and Werner, W. (2012). Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176 1827-1917. · Zbl 1271.60090
[52] Simon, B. (1974). The \(P ( \phi ) _{ 2 }\) Euclidean ( Quantum ) Field Theory . Princeton Univ. Press, Princeton, N.J. · Zbl 1175.81146
[53] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244. · Zbl 0985.60090
[54] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435-1467. · Zbl 1200.82011
[55] Werner, W. (2003). SLEs as boundaries of clusters of Brownian loops. C. R. Math. Acad. Sci. Paris 337 481-486. · Zbl 1029.60085
[56] Werner, W. (2009). Lectures on two-dimensional critical percolation. In Statistical Mechanics. IAS/Park City Math. Ser. 16 297-360. Amer. Math. Soc., Providence, RI. · Zbl 1180.82003
[57] Wu, T. T. (1966). Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I. Phys. Rev. 149 380-401.
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