Abstract
We prove a general result on convergence of interfaces in the critical planar Ising model to conformally invariant curves absolutely continuous with respect to SLE(3). Our setup includes multiple interfaces on arbitrary finitely connected domains, and we also treat the radial SLE case. In the case of simply and doubly connected domains, the limiting processes are described explicitly in terms of rational and elliptic functions, respectively.
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References
Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the Conference Conformal Invariance and Random Spatial Processes, Edinburgh (2003)
Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5), 1125–1163 (2005)
Chelkak, D., Izyurov, K.: Holomorphic spinor observables in the critical Ising model. Commun. Math. Phys. 322(2), 303–332 (2013)
Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)
Chelkak, D., Duminil-Copin, H., Hongler, C.: Crossing probabilities in topological rectangles for the critical planar FK-Ising model (2013). arXiv:1312.7785
Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, Stanislav: Convergence of Ising interfaces to Schramms SLE curves. C. R. Math. 352(2), 157161 (2014)
Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2) 181(3), 1087–1138 (2015)
Dubédat, J.: Commutation relations for Schramm–Loewner evolutions. Commun. Pure Appl. Math. 60(12), 17921847 (2007)
Flores, S., Kleban, P.: A solution space for a system of null-state partial differential equations: Parts I-IV, Commun. Math. Phys. 333(1): 389-481, 333(2): 597–715, (2015)
Hongler, C.: Conformal invariance of Ising model correlations, Ph.D. thesis (2010)
Hongler, C., Kytölä, K.: Ising interfaces and free boundary conditions. J. AMS 26, 1107–1189 (2013)
Izyurov, K.: Holomorphic spinor observables and interfaces in the critical ising model, Ph.D. Thesis. University of Geneva (2011)
Izyurov, K.: Smirnov’s observables for free boundary conditions, interfaces and crossing probabilities. Commun. Math. Phys. 337(1), 225–252 (2015)
Kemppainen, A., Smirnov, S.: Random curves, scaling limits and Loewner evolutions (2012). arXiv:1212.6215
Kemppainen, A., Smirnov, S.: Conformal invariance of boundary touching loops of FK Ising model (2015). arXiv:1509.08858
Kytölä, K., Peltola, E.: Conformally covariant boundary correlation functions with a quantum group (2014). arXiv:1408.1384
Kytölä, K., Peltola, E.: Pure partition functions of multiple SLEs (2015). arXiv:1506.02476
Lawler, G.F.: Defining SLE in multiply connected domains with the Brownian loop measure (2011). arXiv:1108.4364
Lawler, G.F.: Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)
Lawler, G.F., Kozdron, M.: The configuration measure on mutually avoiding SLE paths. In: Fields Institute Communications, vol. 50, pp. 199–224 (2007)
Lawler, G.F., Schramm, O., Werner,W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees, The Annals of Probability, vol. 32, no. 1, B, pp. 939–995 (2004)
Palmer, J.: Planar Ising correlations, Progress inMathematical Physics, vol. 49. Birkhäuser Boston Inc., Boston (2007)
Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)
Reinhardt, W.P., Walker, P.L.: Jacobian elliptic functions. In: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118(1), 221–288 (2000)
Schramm, O., Sheffield, S.: Harmonic explorer and its convergence to \({\rm SLE}_4\). Ann. Probab. 33(6), 2127–2148 (2005)
Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)
Smirnov, S.: Critical percolation in the plane: conformal invariance. Cardy’s formula, scaling limits. Comptes Rendus de l’Acadmie des Sciences I 333(3), 239–244 (2001)
Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich (2006)
Smirnov, S.: Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. Math. (2) 172, 1435–1467 (2010)
Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1840, pp. 107–195. Springer, Berlin (2004)
Zhan, D.: Stochastic Loewner evolution in doubly connected domains. Probab. Theory Related Fields 129–3, 340–380 (2004)
Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008)
Zhan, D.: Reversibility of whole-plane SLE. Probab. Theory Related Fields 0178-8051, 1–58 (2014, online)
Acknowledgments
The author is grateful to the referee for valuable remarks. Work supported by Academy of Finland.
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Izyurov, K. Critical Ising interfaces in multiply-connected domains. Probab. Theory Relat. Fields 167, 379–415 (2017). https://doi.org/10.1007/s00440-015-0685-x
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DOI: https://doi.org/10.1007/s00440-015-0685-x
Mathematics Subject Classification
- 82B20
- 60J67