Ising interfaces and free boundary conditions

Hongler, Clément; Kytölä, Kalle

doi:10.1090/S0894-0347-2013-00774-2

Ising interfaces and free boundary conditions
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by Clément Hongler and Kalle Kytölä;

J. Amer. Math. Soc. 26 (2013), 1107-1189

DOI: https://doi.org/10.1090/S0894-0347-2013-00774-2

Published electronically: June 25, 2013

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Abstract:

We study the interfaces arising in the two-dimensional Ising model at critical temperature, without magnetic field. We show that in the presence of free boundary conditions between plus and minus spins, the scaling limit of these interfaces can be described by a variant of SLE, called dipolar SLE(3). This generalizes a celebrated result of Chelkak and Smirnov and proves a conjecture of Bauer, Bernard, and Houdayer. We mention two possible applications of our result.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, DC, 1964. For sale by the Superintendent of Documents. MR 167642
M. Aizenman and A. Burchard, Hölder regularity and dimension bounds for random curves, Duke Math. J. 99 (1999), no. 3, 419–453. MR 1712629, DOI 10.1215/S0012-7094-99-09914-3
Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989. Reprint of the 1982 original. MR 998375
M. Bauer, D. Bernard, and J. Houdayer, Dipolar stochastic Loewner evolutions, J. Stat. Mech. Theory Exp. 3 (2005), P03001, 18. MR 2140124, DOI 10.1088/1742-5468/2005/03/p03001
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), no. 2, 333–380. MR 757857, DOI 10.1016/0550-3213(84)90052-X
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry of critical fluctuations in two dimensions, J. Statist. Phys. 34 (1984), no. 5-6, 763–774. MR 751712, DOI 10.1007/BF01009438
Dmitri Beliaev and Konstantin Izyurov, A proof of factorization formula for critical percolation, Comm. Math. Phys. 310 (2012), no. 3, 611–623. MR 2891868, DOI 10.1007/s00220-011-1335-5
S. Benoist, J. Dubédat, in preparation.
Federico Camia and Charles M. Newman, Critical percolation exploration path and $\textrm {SLE}_6$: a proof of convergence, Probab. Theory Related Fields 139 (2007), no. 3-4, 473–519. MR 2322705, DOI 10.1007/s00440-006-0049-7
Federico Camia and Charles M. Newman, Two-dimensional critical percolation: the full scaling limit, Comm. Math. Phys. 268 (2006), no. 1, 1–38. MR 2249794, DOI 10.1007/s00220-006-0086-1
J. Cardy, Conformal invariance and Surface Critical Behavior. Nucl. Phys. B 240:514–532, 1984.
John L. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), no. 4, L201–L206. MR 1151081
D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen, and S. Smirnov, Convergence of Ising interfaces to Schramm’s SLEs. Preprint.
Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs, Adv. Math. 228 (2011), no. 3, 1590–1630. MR 2824564, DOI 10.1016/j.aim.2011.06.025
Dmitry Chelkak and Stanislav Smirnov, Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math. 189 (2012), no. 3, 515–580. MR 2957303, DOI 10.1007/s00222-011-0371-2
D. Chelkak and S. Smirnov, preprint.
D. Chelkak, Robust Discrete Complex Analysis: A Toolbox. Preprint: arXiv:1212.6205.
D. Chelkak, H. Duminil-Copin, and C. Hongler, Crossing probabilities in topological rectangles for critical planar FK-Ising model. In preparation.
Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR 1424041, DOI 10.1007/978-1-4612-2256-9
Hugo Duminil-Copin, Clément Hongler, and Pierre Nolin, Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model, Comm. Pure Appl. Math. 64 (2011), no. 9, 1165–1198. MR 2839298, DOI 10.1002/cpa.20370
Michael E. Fisher, Renormalization group theory: its basis and formulation in statistical physics, Rev. Modern Phys. 70 (1998), no. 2, 653–681. MR 1627528, DOI 10.1103/RevModPhys.70.653
Christophe Garban, Steffen Rohde, and Oded Schramm, Continuity of the SLE trace in simply connected domains, Israel J. Math. 187 (2012), 23–36. MR 2891697, DOI 10.1007/s11856-011-0161-y
Geoffrey Grimmett, The random-cluster model, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin, 2006. MR 2243761, DOI 10.1007/978-3-540-32891-9
C. Hongler, Conformal invariance of Ising model correlations. Ph.D. thesis, University of Geneva, http://www.math.columbia.edu/~hongler/thesis.pdf, 2010.
C. Hongler and S. Smirnov, The energy density in the critical planar Ising model. To appear in Acta Math., arXiv:1008.2645.
E. Ising, Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31:253–258, 1925.
Leo P. Kadanoff and Horacio Ceva, Determination of an operator algebra for the two-dimensional Ising model, Phys. Rev. B (3) 3 (1971), 3918–3939. MR 389111
Antti Kemppainen, Stationarity of SLE, J. Stat. Phys. 139 (2010), no. 1, 108–121. MR 2602985, DOI 10.1007/s10955-010-9929-4
A. Kemppainen and S. Smirnov, Random curves, scaling limits and Loewner evolutions. Preprint: arXiv:1212.6215, 2012.
A. Kemppainen and S. Smirnov, in preparation.
Richard Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, 759–795. MR 1782431, DOI 10.1214/aop/1019160260
Kalle Kytölä, On conformal field theory of $\textrm {SLE}(\kappa ,\rho )$, J. Stat. Phys. 123 (2006), no. 6, 1169–1181. MR 2253874, DOI 10.1007/s10955-006-9138-3
K. Kytölä, Conformal field theory methods for variants of Schramm Loewner evolutions, Ph.D, thesis, 2006.
H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. I, Phys. Rev. (2) 60 (1941), 252–262. MR 4803
Robert P. Langlands, Marc-André Lewis, and Yvan Saint-Aubin, Universality and conformal invariance for the Ising model in domains with boundary, J. Statist. Phys. 98 (2000), no. 1-2, 131–244. MR 1745839, DOI 10.1023/A:1018674822185
Robert Langlands, Philippe Pouliot, and Yvan Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1–61. MR 1230963, DOI 10.1090/S0273-0979-1994-00456-2
Gregory F. Lawler, Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR 2129588, DOI 10.1090/surv/114
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001), no. 2, 275–308. MR 1879851, DOI 10.1007/BF02392619
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939–995. MR 2044671, DOI 10.1214/aop/1079021469
W. Lenz, Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. Phys. Zeitschr., 21:613–615, 1920.
Nikolai Makarov and Stanislav Smirnov, Off-critical lattice models and massive SLEs, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 362–371. MR 2730811, DOI 10.1142/9789814304634_{0}024
B. M. McCoy and T. T. Wu, The two-dimensional Ising model. Harvard University Press, Cambridge, Massachusetts, 1973.
J. Miller, Universality for SLE$\left (4\right )$. Preprint: arXiv:1010.1356.
Lars Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. (2) 65 (1944), 117–149. MR 10315
John Palmer, Planar Ising correlations, Progress in Mathematical Physics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2332010
R. Peierls, On Ising’s model of ferromagnetism. Proc. Camb. Philos. Soc., 32:477–481, 1936.
C.-E. Pfister and Y. Velenik, Interface, surface tension and reentrant pinning transition in the $2$D Ising model, Comm. Math. Phys. 204 (1999), no. 2, 269–312. MR 1704276, DOI 10.1007/s002200050646
C. Pommerenke, Boundary Behavior of Conformal Maps, A Series of Comprehensive Studies in Mathematics 299. Springer-Verlag, Berlin, 1992.
Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883–924. MR 2153402, DOI 10.4007/annals.2005.161.883
Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, DOI 10.1007/BF02803524
Oded Schramm, Conformally invariant scaling limits: an overview and a collection of problems, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 513–543. MR 2334202, DOI 10.4171/022-1/20
Oded Schramm and Scott Sheffield, Contour lines of the two-dimensional discrete Gaussian free field, Acta Math. 202 (2009), no. 1, 21–137. MR 2486487, DOI 10.1007/s11511-009-0034-y
Oded Schramm and David B. Wilson, SLE coordinate changes, New York J. Math. 11 (2005), 659–669. MR 2188260
Scott Sheffield, Exploration trees and conformal loop ensembles, Duke Math. J. 147 (2009), no. 1, 79–129. MR 2494457, DOI 10.1215/00127094-2009-007
Scott Sheffield and Wendelin Werner, Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. of Math. (2) 176 (2012), no. 3, 1827–1917. MR 2979861, DOI 10.4007/annals.2012.176.3.8
Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244 (English, with English and French summaries). MR 1851632, DOI 10.1016/S0764-4442(01)01991-7
Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244 (English, with English and French summaries). MR 1851632, DOI 10.1016/S0764-4442(01)01991-7
Stanislav Smirnov, Towards conformal invariance of 2D lattice models, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421–1451. MR 2275653
Stanislav Smirnov, Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. of Math. (2) 172 (2010), no. 2, 1435–1467. MR 2680496, DOI 10.4007/annals.2010.172.1441
Stanislav Smirnov, Discrete complex analysis and probability, Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi, 2010, pp. 595–621. MR 2827906
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423–439. MR 177430, DOI 10.1214/aoms/1177700153
Wendelin Werner, Girsanov’s transformation for $\textrm {SLE}(\kappa ,\rho )$ processes, intersection exponents and hiding exponents, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 1, 121–147 (English, with English and French summaries). MR 2060031
Wendelin Werner, Percolation et modèle d’Ising, Cours Spécialisés [Specialized Courses], vol. 16, Société Mathématique de France, Paris, 2009 (French). MR 2560997
D. Zhan, Random Loewner chains in Riemann surfaces. PhD dissertation, Caltech, 2004.