On multiple SLE for the FK-Ising model. (English) Zbl 1486.60106
Summary: We prove the convergence of multiple interfaces in the critical planar \(q=2\) random cluster model and provide an explicit description of the scaling limit. Remarkably, the expression for the partition function of the resulting multiple \(\mathrm{SLE}_{16/3}\) coincides with the bulk spin correlation in the critical Ising model in the half-plane, after formally replacing a position of each spin and its complex conjugate with a pair of points on the real line. As a corollary we recover Belavin-Polyakov-Zamolodchikov equations for the spin correlations.
MSC:
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
References:
| [1] | BAUER, M., BERNARD, D. and KYTÖLÄ, K. (2005). Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120 1125-1163. · Zbl 1094.82016 · doi:10.1007/s10955-005-7002-5 |
| [2] | BEFFARA, V. and DUMINIL-COPIN, H. (2012). The self-dual point of the two-dimensional random-cluster model is critical for \[q\ge 1\]. Probab. Theory Related Fields 153 511-542. · Zbl 1257.82014 · doi:10.1007/s00440-011-0353-8 |
| [3] | BEFFARA, V., PELTOLA, E. and WU, H. (2021). On the uniqueness of global multiple SLEs. Ann. Probab. 49 400-434. · Zbl 1478.60225 · doi:10.1214/20-AOP1477 |
| [4] | 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 · doi:10.1016/0550-3213(84)90052-X |
| [5] | BURKHARDT, T. W. and GUIM, I. (1987). Bulk, surface, and interface properties of the Ising model and conformal invariance. Phys. Rev. B 36 2080-2083. · doi:10.1103/PhysRevB.36.2080 |
| [6] | CHELKAK, D. (2018). Planar Ising model at criticality: State-of-the-art and perspectives. In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited Lectures 2801-2828. World Sci. Publ., Hackensack, NJ. · Zbl 1453.82006 |
| [7] | CHELKAK, D. (2020). Ising model and s-embeddings of planar graphs. arXiv:2006.14559. |
| [8] | CHELKAK, D., CIMASONI, D. and KASSEL, A. (2017). Revisiting the combinatorics of the 2D Ising model. Ann. Inst. Henri Poincaré D 4 309-385. · Zbl 1380.82017 · doi:10.4171/AIHPD/42 |
| [9] | 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 · doi:10.1214/16-EJP3452 |
| [10] | 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. Paris 352 157-161. · Zbl 1294.82007 · doi:10.1016/j.crma.2013.12.002 |
| [11] | CHELKAK, D., HONGLER, C. and IZYUROV, K. (2015). Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 1087-1138. · Zbl 1318.82006 · doi:10.4007/annals.2015.181.3.5 |
| [12] | CHELKAK, D., HONGLER, C. and IZYUROV, K. (2021). Correlations of primary fields in the critical Ising model. arXiv:2103.10263. · Zbl 1318.82006 |
| [13] | CHELKAK, D., LASLIER, B. and RUSSKIKH, M. (2020). Dimer model and holomorphic functions on t-embeddings of planar graphs. arXiv:2001.11871. |
| [14] | 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 · doi:10.1007/s00222-011-0371-2 |
| [15] | DUBÉDAT, J. (2007). Commutation relations for Schramm-Loewner evolutions. Comm. Pure Appl. Math. 60 1792-1847. · Zbl 1137.82009 · doi:10.1002/cpa.20191 |
| [16] | DUBÉDAT, J. (2009). SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 995-1054. · Zbl 1204.60079 · doi:10.1090/S0894-0347-09-00636-5 |
| [17] | 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 · doi:10.1002/cpa.20370 |
| [18] | DUMINIL-COPIN, H. and SMIRNOV, S. (2012). Conformal invariance of lattice models. In Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc. 15 213-276. Amer. Math. Soc., Providence, RI. · Zbl 1253.82012 · doi:10.4007/annals.2012.175.3.14 |
| [19] | EDWARDS, R. G. and SOKAL, A. D. (1988). Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38 2009-2012. · doi:10.1103/PhysRevD.38.2009 |
| [20] | FLORES, S. M. and KLEBAN, P. (2015). A solution space for a system of null-state partial differential equations: Part 1. Comm. Math. Phys. 333 389-434. · Zbl 1314.35188 · doi:10.1007/s00220-014-2189-4 |
| [21] | FLORES, S. M. and KLEBAN, P. (2015). A solution space for a system of null-state partial differential equations: Part 2. Comm. Math. Phys. 333 435-481. · Zbl 1314.35189 · doi:10.1007/s00220-014-2185-8 |
| [22] | FLORES, S. M. and KLEBAN, P. (2015). A solution space for a system of null-state partial differential equations: Part 3. Comm. Math. Phys. 333 597-667. · Zbl 1311.35314 · doi:10.1007/s00220-014-2190-y |
| [23] | FLORES, S. M. and KLEBAN, P. (2015). A solution space for a system of null-state partial differential equations: Part 4. Comm. Math. Phys. 333 669-715. · Zbl 1314.35190 · doi:10.1007/s00220-014-2180-0 |
| [24] | FOMIN, S. (2001). Loop-erased walks and total positivity. Trans. Amer. Math. Soc. 353 3563-3583. · Zbl 0973.15014 · doi:10.1090/S0002-9947-01-02824-0 |
| [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] | GRAHAM, K. (2007). On multiple Schramm-Loewner evolutions. J. Stat. Mech. Theory Exp. 3 P03008, 21. · Zbl 1456.60213 · doi:10.1088/1742-5468/2007/03/p03008 |
| [27] | GRIMMETT, G. (2004). The random-cluster model. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 73-123. Springer, Berlin. · Zbl 1045.60105 · doi:10.1007/978-3-662-09444-0_2 |
| [28] | HAGENDORF, C., BERNARD, D. and BAUER, M. (2010). The Gaussian free field and \[{\text{SLE}_4}\] on doubly connected domains. J. Stat. Phys. 140 1-26. · Zbl 1193.82027 · doi:10.1007/s10955-010-9980-1 |
| [29] | HONGLER, C. and KYTÖLÄ, K. (2013). Ising interfaces and free boundary conditions. J. Amer. Math. Soc. 26 1107-1189. · Zbl 1284.82021 · doi:10.1090/S0894-0347-2013-00774-2 |
| [30] | IZYUROV, K. (2015). Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities. Comm. Math. Phys. 337 225-252. · Zbl 1318.82010 · doi:10.1007/s00220-015-2339-3 |
| [31] | IZYUROV, K. (2017). Critical Ising interfaces in multiply-connected domains. Probab. Theory Related Fields 167 379-415. · Zbl 1364.82012 · doi:10.1007/s00440-015-0685-x |
| [32] | IZYUROV, K. and KYTÖLÄ, K. (2013). Hadamard’s formula and couplings of SLEs with free field. Probab. Theory Related Fields 155 35-69. · Zbl 1269.60067 · doi:10.1007/s00440-011-0391-2 |
| [33] | KADANOFF, L. P. and CEVA, H. (1971). Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B 3 3918-3939. |
| [34] | KARRILA, A. (2018). Limits of conformal images and conformal images of limits for planar random curves. arXiv:1810.05608. |
| [35] | KARRILA, A. (2019). Multiple SLE type scaling limits: From local to global. arXiv:1903.10354. |
| [36] | KARRILA, A. (2020). UST branches, martingales, and multiple \[\text{SLE}(2)\]. Electron. J. Probab. 25 Paper No. 83, 37. · Zbl 1459.60175 · doi:10.1214/20-ejp485 |
| [37] | KARRILA, A., KYTÖLÄ, K. and PELTOLA, E. (2019). Conformal blocks, \(q\)-combinatorics, and quantum group symmetry. Ann. Inst. Henri Poincaré D 6 449-487. · Zbl 1432.81054 · doi:10.4171/AIHPD/88 |
| [38] | KARRILA, A., KYTÖLÄ, K. and PELTOLA, E. (2020). Boundary correlations in planar LERW and UST. Comm. Math. Phys. 376 2065-2145. · Zbl 1441.82023 · doi:10.1007/s00220-019-03615-0 |
| [39] | KEMPPAINEN, A. (2017). Schramm-Loewner Evolution. SpringerBriefs in Mathematical Physics 24. Springer, Cham. · Zbl 1422.60004 · doi:10.1007/978-3-319-65329-7 |
| [40] | KEMPPAINEN, A. and SMIRNOV, S. (2016). Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527. |
| [41] | KEMPPAINEN, A. and SMIRNOV, S. (2017). Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45 698-779. · Zbl 1393.60016 · doi:10.1214/15-AOP1074 |
| [42] | KEMPPAINEN, A. and SMIRNOV, S. (2018). Configurations of FK Ising interfaces and hypergeometric SLE. Math. Res. Lett. 25 875-889. · Zbl 1418.82002 · doi:10.4310/MRL.2018.v25.n3.a7 |
| [43] | KEMPPAINEN, A. and SMIRNOV, S. (2019). Conformal invariance of boundary touching loops of FK Ising model. Comm. Math. Phys. 369 49-98. · Zbl 1422.60140 · doi:10.1007/s00220-019-03437-0 |
| [44] | KEMPPAINEN, A. and TUISKU, P. (2020). In preparation. |
| [45] | KOZDRON, M. J. and LAWLER, G. F. (2005). Estimates of random walk exit probabilities and application to loop-erased random walk. Electron. J. Probab. 10 1442-1467. · Zbl 1110.60046 · doi:10.1214/EJP.v10-294 |
| [46] | KYTÖLÄ, K. and PELTOLA, E. (2016). Pure partition functions of multiple SLEs. Comm. Math. Phys. 346 237-292. · Zbl 1358.82012 · doi:10.1007/s00220-016-2655-2 |
| [47] | KYTÖLÄ, K. and PELTOLA, E. (2020). Conformally covariant boundary correlation functions with a quantum group. J. Eur. Math. Soc. (JEMS) 22 55-118. · Zbl 1454.81112 · doi:10.4171/jems/917 |
| [48] | Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI. · Zbl 1074.60002 · doi:10.1090/surv/114 |
| [49] | 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 · doi:10.1214/aop/1079021469 |
| [50] | PELTOLA, E. and WU, H. (2018). Crossing probabilities of multiple Ising interfaces. arXiv:1808.09438. |
| [51] | PELTOLA, E. and WU, H. (2019). Global and local multiple SLEs for \[\kappa \le 4\] and connection probabilities for level lines of GFF. Comm. Math. Phys. 366 469-536. · Zbl 1422.60142 · doi:10.1007/s00220-019-03360-4 |
| [52] | PROTTER, P. E. (2004). Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin. · Zbl 1041.60005 |
| [53] | Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093 · doi:10.1007/BF02803524 |
| [54] | SCHRAMM, O. and SHEFFIELD, S. (2005). Harmonic explorer and its convergence to \[{\text{SLE}_4} \]. Ann. Probab. 33 2127-2148. · Zbl 1095.60007 · doi:10.1214/009117905000000477 |
| [55] | Schramm, O. and Sheffield, S. (2013). A contour line of the continuum Gaussian free field. Probab. Theory Related Fields 157 47-80. · Zbl 1331.60090 · doi:10.1007/s00440-012-0449-9 |
| [56] | SIMMONS, J. J. H., KLEBAN, P., FLORES, S. M. and ZIFF, R. M. (2011). Cluster densities at 2D critical points in rectangular geometries. J. Phys. A 44 385002, 34. · Zbl 1227.82033 · doi:10.1088/1751-8113/44/38/385002 |
| [57] | SMIRNOV, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421-1451. Eur. Math. Soc., Zürich. · Zbl 1112.82014 |
| [58] | 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 · doi:10.4007/annals.2010.172.1441 |
| [59] | 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 · doi:10.1090/pcms/016/06 |
| [60] | Zhan, D. (2008). Duality of chordal SLE. Invent. Math. 174 309-353. · Zbl 1158.60047 · doi:10.1007/s00222-008-0132-z |
| [61] | ZHAN, D. (2008). The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36 467-529 · Zbl 1153.60057 · doi:10.1214/07-AOP342 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
