Comptes Rendus
Probability theory/Mathematical physics
Convergence of Ising interfaces to Schrammʼs SLE curves
[Convergence des interfaces dʼIsing vers les courbes SLE introduites par Schramm]
Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 157-161.

We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schrammʼs SLE curves with parameters κ=3 and κ=16/3, respectively.

Cet article explique comment combiner certains résultats antérieurs des différents auteurs afin de montrer la convergence forte des interfaces du modèle dʼIsing critique planaire et de sa représentation FK vers les courbes SLE(3) et SLE(16/3) introduites par Schramm.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.12.002

Dmitry Chelkak 1, 2 ; Hugo Duminil-Copin 3 ; Clément Hongler 4 ; Antti Kemppainen 5 ; Stanislav Smirnov 1, 3

1 Chebyshev Laboratory, Department of Mathematics and Mechanics, St. Petersburg State University, Russian Federation
2 St. Petersburg Department of Steklov Mathematical Institute (PDMI RAS), Russian Federation
3 Section de Mathématiques, Université de Genève, Switzerland
4 Department of Mathematics, Columbia University, United States
5 Department of Mathematics and Statistics, University of Helsinki, Finland
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Dmitry Chelkak; Hugo Duminil-Copin; Clément Hongler; Antti Kemppainen; Stanislav Smirnov. Convergence of Ising interfaces to Schrammʼs SLE curves. Comptes Rendus. Mathématique, Volume 352 (2014) no. 2, pp. 157-161. doi : 10.1016/j.crma.2013.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.12.002/

[1] M. Aizenman; A. Burchard Hölder regularity and dimension bounds for random curves, Duke Math. J., Volume 99 (1999) no. 3, pp. 419-453

[2] A.A. Belavin; A.M. Polyakov; A.B. Zamolodchikov Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys., Volume 34 (1984) no. 5–6, pp. 763-774

[3] D. Chelkak Robust discrete complex analysis: a toolbox, 2012 (Preprint) | arXiv

[4] D. Chelkak, H. Duminil-Copin, Clément Hongler, Crossing probabilities in topological rectangles for the critical planar FK Ising model, Preprint, , 2013. | arXiv

[5] D. Chelkak; C. Hongler; K. Izyurov Conformal invariance of spin correlations in the planar Ising model, 2012 (Preprint) | arXiv

[6] D. Chelkak; K. Izyurov Holomorphic spinor observables in the critical Ising model, Commun. Math. Phys., Volume 322 (2013) no. 2, pp. 303-332

[7] D. Chelkak; S. Smirnov Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., Volume 189 (2012) no. 3, pp. 515-580

[8] H. Duminil-Copin; C. Hongler; P. Nolin Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model, Commun. Pure Appl. Math., Volume 64 (2011) no. 9, pp. 1165-1198

[9] H. Duminil-Copin; S. Smirnov Conformal invariance of lattice models, Probability and Statistical Physics in Two and More Dimensions, Clay Math. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 2012, pp. 213-276

[10] C. Hongler Conformal invariance of Ising model correlations, 2010 (PhD thesis)

[11] C. Hongler; S. Smirnov The energy density in the planar Ising model, Acta. Math., Volume 211 (2013) no. 2, pp. 191-225

[12] A. Kemppainen; S. Smirnov Random curves, scaling limits and Loewner evolutions, 2012 (Preprint) | arXiv

[13] G.F. Lawler Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005

[14] G.F. Lawler; O. Schramm; W. Werner Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., Volume 32 (2004) no. 1B, pp. 939-995

[15] C. Pommerenke Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 299, Springer-Verlag, Berlin, 1992

[16] O. Schramm Scaling limits of loop-erased random walks and uniform spanning trees, Isr. J. Math., Volume 118 (2000), pp. 221-288

[17] S. Smirnov Towards conformal invariance of 2D lattice models, International Congress of Mathematicians, vol. II, Eur. Math. Soc., Zurich, 2006, pp. 1421-1451

[18] S. Smirnov Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. Math., Volume 172 (2010) no. 2, pp. 1435-1467

  • Yong Han; Mingchang Liu; Hao Wu Hypergeometric SLE with κ=8: Convergence of UST and LERW in topological rectangles, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 61 (2025) no. 2 | DOI:10.1214/23-aihp1446
  • Alexander Glazman; Piet Lammers Delocalisation and continuity in 2D: loop O(2), six-vertex, and random-cluster models, Communications in Mathematical Physics, Volume 406 (2025) no. 5, p. 59 (Id/No 108) | DOI:10.1007/s00220-025-05259-9 | Zbl:8030971
  • Yan Luo; Sid Maibach Two-Loop Loewner Potentials, International Mathematics Research Notices, Volume 2025 (2025) no. 11 | DOI:10.1093/imrn/rnaf133
  • Giulia Cava; Alessandro Giuliani; Rafael Leon Greenblatt The scaling limit of boundary spin correlations in non-integrable Ising models, Journal of Mathematical Physics, Volume 66 (2025) no. 2, p. 41 (Id/No 023301) | DOI:10.1063/5.0235381 | Zbl:7993802
  • Laurin Köhler-Schindler; Matthis Lehmkuehler The fuzzy Potts model in the plane: scaling limits and arm exponents, Probability Theory and Related Fields, Volume 191 (2025) no. 1-2, pp. 287-359 | DOI:10.1007/s00440-024-01319-8 | Zbl:7990325
  • Yusuke Shibasaki On the Role of Loewner Entropy in the Statistical Mechanics of the 2D Ising System, Progress of Theoretical and Experimental Physics, Volume 2025 (2025) no. 2 | DOI:10.1093/ptep/ptaf017
  • Federico Camia; Yu Feng Conformal covariance of connection probabilities in the 2D critical FK-Ising model, Stochastic Processes and their Applications, Volume 189 (2025), p. 104734 | DOI:10.1016/j.spa.2025.104734
  • Nicholas Crawford; Alexander Glazman; Matan Harel; Ron Peled Macroscopic loops in the loop O(n) model via the XOR trick, The Annals of Probability, Volume 53 (2025) no. 2, pp. 478-508 | DOI:10.1214/24-aop1712 | Zbl:8050989
  • Morris Ang; Nina Holden; Xin Sun Integrability of SLE via conformal welding of random surfaces, Communications on Pure and Applied Mathematics, Volume 77 (2024) no. 5, pp. 2651-2707 | DOI:10.1002/cpa.22180 | Zbl:1543.81066
  • Xin Sun; Pu Yu SLE partition functions via conformal welding of random surfaces, IMRN. International Mathematics Research Notices, Volume 2024 (2024) no. 24, pp. 14763-14801 | DOI:10.1093/imrn/rnae260 | Zbl:7986123
  • Yu Feng; Hao Wu; Lu Yang Multiple Ising Interfaces in Annulus and 2N-Sided Radial SLE, International Mathematics Research Notices, Volume 2024 (2024) no. 6, p. 5326 | DOI:10.1093/imrn/rnad252
  • Claude Godrèche; Marco Picco Interfaces of the two-dimensional voter model in the context of SLE, Journal of Statistical Mechanics: Theory and Experiment, Volume 2024 (2024) no. 11, p. 113203 | DOI:10.1088/1742-5468/ad8225
  • Alex M. Karrila Limits of conformal images and conformal images of limits for planar random curves, L'Enseignement Mathématique. 2e Série, Volume 70 (2024) no. 3-4, pp. 385-423 | DOI:10.4171/lem/1066 | Zbl:1548.60041
  • J. Amette Estrada; M. Noseda; P. J. Cobelli; P. D. Mininni Conformal invariance in out-of-equilibrium Bose-Einstein condensates governed by the Gross-Pitaevskii equation, Physical Review A, Volume 109 (2024) no. 6 | DOI:10.1103/physreva.109.l061302
  • M. Noseda; P. J. Cobelli Conformal Invariance in Water-Wave Turbulence, Physical Review Letters, Volume 132 (2024) no. 9 | DOI:10.1103/physrevlett.132.094001
  • Yu Feng; Eveliina Peltola; Hao Wu Connection probabilities of multiple FK-Ising interfaces, Probability Theory and Related Fields, Volume 189 (2024) no. 1-2, pp. 281-367 | DOI:10.1007/s00440-024-01269-1 | Zbl:1541.82002
  • Yiting Li; Xin Sun; Samuel S. Watson Schnyder woods, SLE16, and Liouville quantum gravity, Transactions of the American Mathematical Society, Volume 377 (2024) no. 4, pp. 2439-2493 | DOI:10.1090/tran/8887 | Zbl:1543.60097
  • Sung-Soo Byun; Nam-Gyu Kang; Hee-Joon Tak Conformal field theory for annulus SLE: partition functions and martingale-observables, Analysis and Mathematical Physics, Volume 13 (2023) no. 1, p. 87 (Id/No 1) | DOI:10.1007/s13324-022-00761-y | Zbl:7624354
  • Giovanni Antinucci; Alessandro Giuliani; Rafael L. Greenblatt Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, Communications in Mathematical Physics, Volume 397 (2023) no. 1, pp. 393-483 | DOI:10.1007/s00220-022-04481-z | Zbl:7676189
  • Scott Sheffield What is a random surface?, International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 2. Plenary lectures, Berlin: European Mathematical Society (EMS), 2023, pp. 1202-1258 | DOI:10.4171/icm2022/187 | Zbl:1533.60076
  • Alessandro Giuliani Scaling limits and universality of Ising and dimer models, International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 5. Sections 9–11, Berlin: European Mathematical Society (EMS), 2023, pp. 4040-4072 | DOI:10.4171/icm2022/131 | Zbl:1535.82014
  • Olivier Bernardi; Nina Holden; Xin Sun Percolation on triangulations: a bijective path to Liouville quantum gravity, Memoirs of the American Mathematical Society, 1440, Providence, RI: American Mathematical Society (AMS), 2023 | DOI:10.1090/memo/1440 | Zbl:7753145
  • Eveliina Peltola; Hao Wu Crossing probabilities of multiple Ising interfaces, The Annals of Applied Probability, Volume 33 (2023) no. 4, pp. 3169-3206 | DOI:10.1214/22-aap1888 | Zbl:1520.82015
  • Dmitry Chelkak; Konstantin Izyurov; Rémy Mahfouf Universality of spin correlations in the Ising model on isoradial graphs, The Annals of Probability, Volume 51 (2023) no. 3, pp. 840-898 | DOI:10.1214/22-aop1595 | Zbl:1517.82012
  • Rajarshi Mukherjee; Gourab Ray On testing for parameters in Ising models, Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, Volume 58 (2022) no. 1, pp. 164-187 | DOI:10.1214/21-aihp1157 | Zbl:1493.62039
  • S. C. Park Convergence of fermionic observables in the massive planar FK-Ising model, Communications in Mathematical Physics, Volume 396 (2022) no. 3, pp. 1071-1133 | DOI:10.1007/s00220-022-04488-6 | Zbl:1504.82012
  • Erik Bates; Shirshendu Ganguly; Alan Hammond Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape, Electronic Journal of Probability, Volume 27 (2022), p. 44 (Id/No 1) | DOI:10.1214/21-ejp706 | Zbl:1496.60115
  • Hugo Duminil-Copin; Ioan Manolescu Planar random-cluster model: scaling relations, Forum of Mathematics, Pi, Volume 10 (2022), p. 83 (Id/No e23) | DOI:10.1017/fmp.2022.16 | Zbl:1512.60065
  • Taha Ameen; Kalle Kytölä; S. C. Park; David Radnell Slit-strip Ising boundary conformal field theory. I: Discrete and continuous function spaces, Mathematical Physics, Analysis and Geometry, Volume 25 (2022) no. 4, p. 53 (Id/No 30) | DOI:10.1007/s11040-022-09442-5 | Zbl:1508.30085
  • Yusuke Shibasaki Permutation-Loewner entropy analysis for 2-dimensional Ising system interface, Physica A: Statistical Mechanics and its Applications, Volume 594 (2022), p. 126943 | DOI:10.1016/j.physa.2022.126943
  • Pavel Galashin A formula for boundary correlations of the critical Ising model, Probability Theory and Related Fields, Volume 182 (2022) no. 1-2, pp. 615-640 | DOI:10.1007/s00440-021-01086-w | Zbl:1482.82021
  • Adam Bowditch; Rongfeng Sun The two-dimensional continuum random field Ising model, The Annals of Probability, Volume 50 (2022) no. 2, pp. 419-454 | DOI:10.1214/21-aop1536 | Zbl:1489.82040
  • Konstantin Izyurov On multiple SLE for the FK-Ising model, The Annals of Probability, Volume 50 (2022) no. 2, pp. 771-790 | DOI:10.1214/21-aop1547 | Zbl:1486.60106
  • Alexander Glazman; Ioan Manolescu Uniform Lipschitz functions on the triangular lattice have logarithmic variations, Communications in Mathematical Physics, Volume 381 (2021) no. 3, pp. 1153-1221 | DOI:10.1007/s00220-020-03920-z | Zbl:1470.60278
  • Ilia Binder Rate of convergence of critical interfaces to SLE curves, Extended abstracts fall 2019. Spaces of analytic functions: approximation, interpolation, sampling, SAFAIS 2019 program, Centre de Recerca Matemàtica, Barcelona, Spain, October–December 2019, Cham: Birkhäuser, 2021, pp. 43-50 | DOI:10.1007/978-3-030-74417-5_7 | Zbl:1504.30001
  • Makoto Katori; Shinji Koshida Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions, Journal of Physics A: Mathematical and Theoretical, Volume 54 (2021) no. 32, p. 19 (Id/No 325002) | DOI:10.1088/1751-8121/ac0dee | Zbl:1519.60093
  • Hugo Duminil-Copin; Alexander Glazman; Ron Peled; Yinon Spinka Macroscopic loops in the loop O(n) model at Nienhuis' critical point, Journal of the European Mathematical Society (JEMS), Volume 23 (2021) no. 1, pp. 315-347 | DOI:10.4171/jems/1012 | Zbl:1477.60137
  • Hugo Duminil-Copin; Ioan Manolescu; Vincent Tassion Planar random-cluster model: fractal properties of the critical phase, Probability Theory and Related Fields, Volume 181 (2021) no. 1-3, pp. 401-449 | DOI:10.1007/s00440-021-01060-6 | Zbl:1475.60195
  • Yusuke Shibasaki Permutation-Loewner Entropy Analysis for 2-Dimensional Ising System Interface, SSRN Electronic Journal (2021) | DOI:10.2139/ssrn.3989699
  • Vincent Beffara; Eveliina Peltola; Hao Wu On the uniqueness of global multiple SLEs, The Annals of Probability, Volume 49 (2021) no. 1, pp. 400-434 | DOI:10.1214/20-aop1477 | Zbl:1478.60225
  • Yusuke Shibasaki; Minoru Saito Loewner driving force of the interface in the 2-dimensional Ising system as a chaotic dynamical system, Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 30 (2020) no. 11 | DOI:10.1063/5.0023261
  • Nahid Ghodratipour; Shahin Rouhani The expectation value of the number of loops and the left-passage probability in the double-dimer model, Communications in Mathematical Physics, Volume 373 (2020) no. 1, pp. 357-388 | DOI:10.1007/s00220-019-03620-3 | Zbl:1475.82004
  • Hao Wu Hypergeometric SLE: conformal Markov characterization and applications, Communications in Mathematical Physics, Volume 374 (2020) no. 2, pp. 433-484 | DOI:10.1007/s00220-020-03697-1 | Zbl:1473.82021
  • Linxiao Chen; Joonas Turunen Critical Ising model on random triangulations of the disk: enumeration and local limits, Communications in Mathematical Physics, Volume 374 (2020) no. 3, pp. 1577-1643 | DOI:10.1007/s00220-019-03672-5 | Zbl:1437.82004
  • Pavel Galashin; Pavlo Pylyavskyy Ising model and the positive orthogonal Grassmannian, Duke Mathematical Journal, Volume 169 (2020) no. 10, pp. 1877-1942 | DOI:10.1215/00127094-2019-0086 | Zbl:1446.82008
  • Huy Tran; Yizheng Yuan A support theorem for SLE curves, Electronic Journal of Probability, Volume 25 (2020), p. 18 (Id/No 18) | DOI:10.1214/20-ejp425 | Zbl:1439.30021
  • Alex Karrila UST branches, martingales, and multiple SLE(2), Electronic Journal of Probability, Volume 25 (2020), p. 37 (Id/No 83) | DOI:10.1214/20-ejp485 | Zbl:1459.60175
  • Makoto Katori; Shinji Koshida Conformal welding problem, flow line problem, and multiple Schramm-Loewner evolution, Journal of Mathematical Physics, Volume 61 (2020) no. 8, p. 083301 | DOI:10.1063/1.5145357 | Zbl:1454.81194
  • Christophe Garban; Hao Wu On the convergence of FK-Ising percolation to SLE16/3(16/36), Journal of Theoretical Probability, Volume 33 (2020) no. 2, pp. 828-865 | DOI:10.1007/s10959-019-00950-9 | Zbl:1434.60230
  • Gesualdo Delfino; Walter Selke; Alessio Squarcini Particles, string and interface in the three-dimensional Ising model, Nuclear Physics. B, Volume 958 (2020), p. 11 (Id/No 115139) | DOI:10.1016/j.nuclphysb.2020.115139 | Zbl:1479.82010
  • Hugo Duminil-Copin Lectures on the Ising and Potts models on the hypercubic lattice, Random graphs, phase transitions, and the Gaussian free field. Lecture notes given at the PIMS-CRM summer school in probability, University of British Columbia, Vancouver, Canada, June 5–30, 2017, Cham: Springer, 2020, pp. 35-161 | DOI:10.1007/978-3-030-32011-9_2 | Zbl:1447.82007
  • Eveliina Peltola; Hao Wu Global and local multiple SLEs for κ4 and connection probabilities for level lines of GFF, Communications in Mathematical Physics, Volume 366 (2019) no. 2, pp. 469-536 | DOI:10.1007/s00220-019-03360-4 | Zbl:1422.60142
  • Reza Gheissari; Clément Hongler; S. C. Park Ising model: local spin correlations and conformal invariance, Communications in Mathematical Physics, Volume 367 (2019) no. 3, pp. 771-833 | DOI:10.1007/s00220-019-03312-y | Zbl:1419.82012
  • Antti Kemppainen; Stanislav Smirnov Conformal invariance of boundary touching loops of FK Ising model, Communications in Mathematical Physics, Volume 369 (2019) no. 1, pp. 49-98 | DOI:10.1007/s00220-019-03437-0 | Zbl:1422.60140
  • Roland Bauerschmidt; David C. Brydges; Gordon Slade Spin Systems, Introduction to a Renormalisation Group Method, Volume 2242 (2019), p. 3 | DOI:10.1007/978-981-32-9593-3_1
  • Michael Aizenman; Hugo Duminil-Copin; Vincent Tassion; Simone Warzel Emergent planarity in two-dimensional Ising models with finite-range interactions, Inventiones Mathematicae, Volume 216 (2019) no. 3, pp. 661-743 | DOI:10.1007/s00222-018-00851-4 | Zbl:1417.82022
  • Hugo Duminil-Copin Sharp threshold phenomena in statistical physics, Japanese Journal of Mathematics. 3rd Series, Volume 14 (2019) no. 1, pp. 1-25 | DOI:10.1007/s11537-018-1726-x | Zbl:1420.82003
  • Eveliina Peltola Toward a conformal field theory for Schramm-Loewner evolutions, Journal of Mathematical Physics, Volume 60 (2019) no. 10, p. 103305 | DOI:10.1063/1.5094364 | Zbl:1431.82020
  • Shinji Koshida Schramm-Loewner-evolution-type growth processes corresponding to Wess-Zumino-Witten theories, Letters in Mathematical Physics, Volume 109 (2019) no. 6, pp. 1397-1413 | DOI:10.1007/s11005-018-01150-y | Zbl:1469.60258
  • Jhih-Huang Li Conformal invariance in the FK-representation of the quantum Ising model and convergence of the interface to the SLE16/3, Probability Theory and Related Fields, Volume 173 (2019) no. 1-2, pp. 87-156 | DOI:10.1007/s00440-018-0831-3 | Zbl:1417.82012
  • Gordon Slade Self-avoiding walk, spin systems and renormalization, Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences, Volume 475 (2019) no. 2221, p. 21 (Id/No 20180549) | DOI:10.1098/rspa.2018.0549 | Zbl:1425.82008
  • Kimihiko Fukushima; Kazumitsu Sakai Numerical study on a crossing probability for the four-state Potts model: Logarithmic correction to the finite-size scaling, Progress of Theoretical and Experimental Physics, Volume 2019 (2019) no. 9 | DOI:10.1093/ptep/ptz101
  • Shinji Koshida Coset construction of Virasoro minimal models and coupling of Wess-Zumino-Witten theory with Schramm-Loewner evolution, Reviews in Mathematical Physics, Volume 31 (2019) no. 10, p. 16 (Id/No 1950037) | DOI:10.1142/s0129055x19500375 | Zbl:1432.60078
  • Ron Peled; Yinon Spinka Lectures on the Spin and Loop O(n) Models, Sojourns in Probability Theory and Statistical Physics - I, Volume 298 (2019), p. 246 | DOI:10.1007/978-981-15-0294-1_10
  • Richard Kenyon; Jason Miller; Scott Sheffield; David B. Wilson Bipolar orientations on planar maps and SLE12, The Annals of Probability, Volume 47 (2019) no. 3, pp. 1240-1269 | DOI:10.1214/18-aop1282 | Zbl:1466.60170
  • Stéphane Benoist; Clément Hongler The scaling limit of critical Ising interfaces is CLE3, The Annals of Probability, Volume 47 (2019) no. 4, pp. 2049-2086 | DOI:10.1214/18-aop1301 | Zbl:1467.60061
  • Wei Qian; Wendelin Werner Coupling the Gaussian free fields with free and with zero boundary conditions via common level lines, Communications in Mathematical Physics, Volume 361 (2018) no. 1, pp. 53-80 | DOI:10.1007/s00220-018-3159-z | Zbl:1430.60064
  • Jason Miller; Wendelin Werner Connection probabilities for conformal loop ensembles, Communications in Mathematical Physics, Volume 362 (2018) no. 2, pp. 415-453 | DOI:10.1007/s00220-018-3207-8 | Zbl:1400.60128
  • Nina Holden; Xin Sun SLE as a mating of trees in Euclidean geometry, Communications in Mathematical Physics, Volume 364 (2018) no. 1, pp. 171-201 | DOI:10.1007/s00220-018-3149-1 | Zbl:1408.60073
  • Hugo Duminil-Copin; Jhih-Huang Li; Ioan Manolescu Universality for the random-cluster model on isoradial graphs, Electronic Journal of Probability, Volume 23 (2018), p. 70 (Id/No 96) | DOI:10.1214/18-ejp223 | Zbl:1414.60076
  • Dmitry Chelkak 2D Ising model: correlation functions at criticality via Riemann-type boundary value problems, European congress of mathematics. Proceedings of the 7th ECM (7ECM) congress, Berlin, Germany, July 18–22, 2016, Zürich: European Mathematical Society (EMS), 2018, pp. 235-256 | DOI:10.4171/176-1/10 | Zbl:1402.82005
  • Hugo Duminil-Copin Random currents expansion of the Ising model, European congress of mathematics. Proceedings of the 7th ECM (7ECM) congress, Berlin, Germany, July 18–22, 2016, Zürich: European Mathematical Society (EMS), 2018, pp. 869-889 | DOI:10.4171/176-1/39 | Zbl:1403.82005
  • Shinji Koshida Schramm-Loewner evolution with Lie superalgebra symmetry, International Journal of Modern Physics A, Volume 33 (2018) no. 20, p. 17 (Id/No 1850117) | DOI:10.1142/s0217751x18501178 | Zbl:1393.60091
  • Shinji Koshida Local martingales associated with Schramm-Loewner evolutions with internal symmetry, Journal of Mathematical Physics, Volume 59 (2018) no. 10, p. 101703 | DOI:10.1063/1.5034416 | Zbl:1402.60103
  • Adrien Poncelet Schramm's formula for multiple loop-erased random walks, Journal of Statistical Mechanics: Theory and Experiment, Volume 2018 (2018) no. 10, p. 66 (Id/No 103106) | DOI:10.1088/1742-5468/aae5a6 | Zbl:1457.82158
  • Hao Wu Polychromatic arm exponents for the critical planar FK-Ising model, Journal of Statistical Physics, Volume 170 (2018) no. 6, pp. 1177-1196 | DOI:10.1007/s10955-018-1983-3 | Zbl:1392.82012
  • Hao Wu Alternating arm exponents for the critical planar Ising model, The Annals of Probability, Volume 46 (2018) no. 5, pp. 2863-2907 | DOI:10.1214/17-aop1241 | Zbl:1428.60119
  • Scott Sheffield; Samuel S. Watson; Hao Wu Simple CLE in doubly connected domains, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 53 (2017) no. 2 | DOI:10.1214/15-aihp726
  • Hugo Duminil-Copin; Vladas Sidoravicius; Vincent Tassion Continuity of the phase transition for planar random-cluster and Potts models with 1q4, Communications in Mathematical Physics, Volume 349 (2017) no. 1, pp. 47-107 | DOI:10.1007/s00220-016-2759-8 | Zbl:1357.82011
  • Hugo Duminil-Copin; Ron Peled; Wojciech Samotij; Yinon Spinka Exponential decay of loop lengths in the loop O(n) model with large n, Communications in Mathematical Physics, Volume 349 (2017) no. 3, pp. 777-817 | DOI:10.1007/s00220-016-2815-4 | Zbl:1359.82007
  • Jean-Christophe Mourrat; Hendrik Weber Convergence of the two-dimensional dynamic Ising-Kac model to ϕ24, Communications on Pure and Applied Mathematics, Volume 70 (2017) no. 4, pp. 717-812 | DOI:10.1002/cpa.21655 | Zbl:1364.82013
  • Hao Wu; Dapeng Zhan Boundary arm exponents for SLE, Electronic Journal of Probability, Volume 22 (2017) no. none | DOI:10.1214/17-ejp110
  • Jason Miller; Scott Sheffield; Wendelin Werner CLE percolations, Forum of Mathematics, Pi, Volume 5 (2017), p. 102 (Id/No e4) | DOI:10.1017/fmp.2017.5 | Zbl:1390.60356
  • Jakob E. Björnberg Fermionic observables in the transverse Ising chain, Journal of Mathematical Physics, Volume 58 (2017) no. 5, p. 053302 | DOI:10.1063/1.4982637 | Zbl:1368.82002
  • Richard Kenyon; Jason Miller; Scott Sheffield; David B. Wilson Six-vertex model and Schramm-Loewner evolution, Physical Review E, Volume 95 (2017) no. 5 | DOI:10.1103/physreve.95.052146
  • Konstantin Izyurov Critical Ising interfaces in multiply-connected domains, Probability Theory and Related Fields, Volume 167 (2017) no. 1-2, pp. 379-415 | DOI:10.1007/s00440-015-0685-x | Zbl:1364.82012
  • Jason Miller; Hao Wu Intersections of SLE paths: the double and cut point dimension of SLE, Probability Theory and Related Fields, Volume 167 (2017) no. 1-2, pp. 45-105 | DOI:10.1007/s00440-015-0677-x | Zbl:1408.60074
  • Menglu Wang; Hao Wu Level lines of Gaussian free field. I: Zero-boundary GFF., Stochastic Processes and their Applications, Volume 127 (2017) no. 4, pp. 1045-1124 | DOI:10.1016/j.spa.2016.07.009 | Zbl:1358.60066
  • Antti Kemppainen; Stanislav Smirnov Random curves, scaling limits and Loewner evolutions, The Annals of Probability, Volume 45 (2017) no. 2 | DOI:10.1214/15-aop1074
  • Niko Jokela; Matti Järvinen; Kalle Kytölä SLE boundary visits, Annales Henri Poincaré, Volume 17 (2016) no. 6, pp. 1263-1330 | DOI:10.1007/s00023-015-0452-7 | Zbl:1346.82012
  • Federico Camia; Christophe Garban; Charles M. Newman Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 52 (2016) no. 1 | DOI:10.1214/14-aihp643
  • Stéphane Benoist; Hugo Duminil-Copin; Clément Hongler Conformal invariance of crossing probabilities for the Ising model with free boundary conditions, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 52 (2016) no. 4 | DOI:10.1214/15-aihp698
  • Kalle Kytölä; Eveliina Peltola Pure partition functions of multiple SLEs, Communications in Mathematical Physics, Volume 346 (2016) no. 1, pp. 237-292 | DOI:10.1007/s00220-016-2655-2 | Zbl:1358.82012
  • Federico Camia; Alberto Gandolfi; Matthew Kleban Conformal correlation functions in the Brownian loop soup, Nuclear Physics. B, Volume 902 (2016), pp. 483-507 | DOI:10.1016/j.nuclphysb.2015.11.022 | Zbl:1332.82083
  • Adrien Kassel; David B. Wilson Active spanning trees and Schramm-Loewner evolution, Physical Review E, Volume 93 (2016) no. 6 | DOI:10.1103/physreve.93.062121
  • Hugo Duminil-Copin; Ioan Manolescu The phase transitions of the planar random-cluster and Potts models with q1 are sharp, Probability Theory and Related Fields, Volume 164 (2016) no. 3-4, pp. 865-892 | DOI:10.1007/s00440-015-0621-0 | Zbl:1356.60167
  • Antti Kemppainen; Wendelin Werner The nested simple conformal loop ensembles in the Riemann sphere, Probability Theory and Related Fields, Volume 165 (2016) no. 3-4, pp. 835-866 | DOI:10.1007/s00440-015-0647-3 | Zbl:1352.60117
  • Tim van de Brug; Federico Camia; Marcin Lis Random walk loop soups and conformal loop ensembles, Probability Theory and Related Fields, Volume 166 (2016) no. 1-2, pp. 553-584 | DOI:10.1007/s00440-015-0666-0 | Zbl:1357.60049
  • Vincent Beffara; Hugo Duminil-Copin Critical point and duality in planar lattice models, Probability and statistical physics in St. Petersburg. St. Petersburg School Probability and Statistical Physics, St. Petersburg State University, St. Petersburg, Russia, June 18–29, 2012, Providence, RI: American Mathematical Society (AMS), 2016, pp. 51-98 | Zbl:1391.82006
  • Dmitry Chelkak Robust discrete complex analysis: a toolbox, The Annals of Probability, Volume 44 (2016) no. 1, pp. 628-683 | DOI:10.1214/14-aop985 | Zbl:1347.60050
  • Dmitry Chelkak; Clément Hongler; Konstantin Izyurov Conformal invariance of spin correlations in the planar Ising model, Annals of Mathematics. Second Series, Volume 181 (2015) no. 3, pp. 1087-1138 | DOI:10.4007/annals.2015.181.3.5 | Zbl:1318.82006
  • Konstantin Izyurov Smirnov's observable for free boundary conditions, interfaces and crossing probabilities, Communications in Mathematical Physics, Volume 337 (2015) no. 1, pp. 225-252 | DOI:10.1007/s00220-015-2339-3 | Zbl:1318.82010
  • Alessandro Giuliani; Vieri Mastropietro; Fabio Toninelli Height fluctuations in non-integrable classical dimers, EPL (Europhysics Letters), Volume 109 (2015) no. 6, p. 60004 | DOI:10.1209/0295-5075/109/60004
  • V. Beffara; H. Duminil-Copin; S. Smirnov On the critical parameters of the q4 random-cluster model on isoradial graphs, Journal of Physics A: Mathematical and Theoretical, Volume 48 (2015) no. 48, p. 28 (Id/No 484003) | DOI:10.1088/1751-8113/48/48/484003 | Zbl:1342.82018
  • Mark Holmes; Yevhen Mohylevskyy; Charles M. Newman The voter model chordal interface in two dimensions, Journal of Statistical Physics, Volume 159 (2015) no. 4, pp. 937-957 | DOI:10.1007/s10955-015-1198-9 | Zbl:1328.82024
  • Abbas Ali Saberi Recent advances in percolation theory and its applications, Physics Reports, Volume 578 (2015), pp. 1-32 | DOI:10.1016/j.physrep.2015.03.003 | Zbl:1357.82032
  • Federico Camia; Christophe Garban; Charles M. Newman Planar Ising magnetization field. I: Uniqueness of the critical scaling limit, The Annals of Probability, Volume 43 (2015) no. 2, pp. 528-571 | DOI:10.1214/13-aop881 | Zbl:1332.82012
  • Hugo Duminil-Copin; Christophe Garban; Gábor Pete The near-critical planar FK-Ising model, Communications in Mathematical Physics, Volume 326 (2014) no. 1, pp. 1-35 | DOI:10.1007/s00220-013-1857-0 | Zbl:1286.82003
  • Jason Miller; Nike Sun; David B. Wilson The Hausdorff dimension of the CLE gasket, The Annals of Probability, Volume 42 (2014) no. 4, pp. 1644-1665 | DOI:10.1214/12-aop820 | Zbl:1305.60078
  • Clément Hongler; Kalle Kytölä Ising interfaces and free boundary conditions, Journal of the American Mathematical Society, Volume 26 (2013) no. 4, pp. 1107-1189 | DOI:10.1090/s0894-0347-2013-00774-2 | Zbl:1284.82021
  • Dmitry Chelkak; Stanislav Smirnov Universality in the 2D Ising model and conformal invariance of fermionic observables, Inventiones Mathematicae, Volume 189 (2012) no. 3, pp. 515-580 | DOI:10.1007/s00222-011-0371-2 | Zbl:1257.82020

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