zbMATH — the first resource for mathematics

Alternating arm exponents for the critical planar Ising model. (English) Zbl 1428.60119
Summary: We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions \((\ominus\oplus)\), \((\ominus\mathrm{free})\) and (free free), and the alternating interior arm exponents.

60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
Full Text: DOI Euclid arXiv
[1] Ahlfors, L. V. (2006). Lectures on Quasiconformal Mappings, 2nd ed. University Lecture Series38. Amer. Math. Soc., Providence, RI. · Zbl 1103.30001
[2] Ahlfors, L. V. (2010). Conformal Invariants: Topics in Geometric Function Theory. AMS Chelsea Publishing, Providence, RI. Reprint of the 1973 original, With a foreword by Peter Duren, F. W. Gehring and Brad Osgood. · Zbl 1211.30002
[3] Alberts, T. and Kozdron, M. J. (2008). Intersection probabilities for a chordal SLE path and a semicircle. Electron. Commun. Probab.13 448–460. · Zbl 1187.82034
[4] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab.36 1421–1452. · Zbl 1165.60007
[5] Benoist, S., Duminil-Copin, H. and Hongler, C. (2016). Conformal invariance of crossing probabilities for the Ising model with free boundary conditions. Ann. Inst. Henri Poincaré Probab. Stat.52 1784–1798. · Zbl 1355.60119
[6] Chelkak, D. (2016). Robust discrete complex analysis: A toolbox. Ann. Probab.44 628–683. · Zbl 1347.60050
[7] Chelkak, D., Duminil-Copin, H. and Hongler, C. (2016). Crossing probabilities in topological rectangles for the critical planar FK–Ising model. Electron. J. Probab.21 Paper No. 5, 28. · Zbl 1341.60124
[8] Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A. and Smirnov, S. (2014). Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris352 157–161. · Zbl 1294.82007
[9] Chelkak, D. and Smirnov, S. (2012). Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math.189 515–580. · Zbl 1257.82020
[10] Dubédat, J. (2009). Duality of Schramm–Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42 697–724. · Zbl 1205.60147
[11] Duplantier, B. (2004). Conformal fractal geometry and boundary quantum gravity. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 365–482. Amer. Math. Soc., Providence, RI. · Zbl 1068.60019
[12] Hongler, C. and Kytölä, K. (2013). Ising interfaces and free boundary conditions. J. Amer. Math. Soc.26 1107–1189. · Zbl 1284.82021
[13] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs114. Amer. Math. Soc., Providence, RI. · Zbl 1074.60002
[14] Lawler, G. F. (2015). Minkowski content of the intersection of a Schramm–Loewner evolution (SLE) curve with the real line. J. Math. Soc. Japan67 1631–1669. · Zbl 1362.60069
[15] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math.187 237–273. · Zbl 1005.60097
[16] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab.7 no. 2, 13 pp. (electronic). · Zbl 1015.60091
[17] Miller, J. and Sheffield, S. (2016). Imaginary geometry I: Interacting SLEs. Probab. Theory Related Fields164 553–705. · Zbl 1336.60162
[18] Miller, J. and Wu, H. (2017). Intersections of SLE Paths: The double and cut point dimension of SLE. Probab. Theory Related Fields167 45–105. · Zbl 1408.60074
[19] Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps299. Springer, Berlin. · Zbl 0762.30001
[20] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924. · Zbl 1081.60069
[21] Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math.11 659–669 (electronic). · Zbl 1094.82007
[22] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett.8 729–744. · Zbl 1009.60087
[23] Viklund, F. J. and Lawler, G. F. (2012). Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math.209 265–322. · Zbl 1271.82007
[24] Werner, W. and Wu, H. (2013). From CLE(\(κ\)) to SLE(\(κ,ρ\)). Electron. J. Probab.18 1–20.
[25] Wu, H. (2018). Polychromatic arm exponents for the critical planar FK–Ising model. J. Stat. Phys.170 1177–1196. · Zbl 1392.82012
[26] Wu, H. · Zbl 1386.60286
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.