Boundary Correlations in Planar LERW and UST

  • Alex Karrila
  • Kalle Kytölä
  • Eveliina PeltolaEmail author


We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple \(\mathrm {SLE}_\kappa \) at \(\kappa =2\).



We thank Christian Hagendorf for useful discussions, and in particular for drawing our attention to the results of [KW11a, KW11b, KW15]. We also thank Dmitry Chelkak, Steven Flores, Christophe Garban, Konstantin Izyurov, Richard Kenyon, Marcin Lis, Wei Qian, David Radnell, Fredrik Viklund, David Wilson, and Hao Wu for interesting and helpful discussions. A.K. and K.K. are supported by the Academy of Finland project “Algebraic structures and random geometry of stochastic lattice models”. During this work, E.P. was supported by Vilho, Yrjö and Kalle Väisälä Foundation and later by the ERC AG COMPASP, the NCCR SwissMAP, and the Swiss NSF.


  1. [BB03]
    Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the Conference ‘Conformal Invariance and Random Spatial Processes’, Edinburgh (2003)Google Scholar
  2. [BBK05]
    Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6), 1125–1163 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [BPW18]
    Beffara, V., Peltola, E., Wu, H.: On the uniqueness of global multiple SLEs. Preprint. arXiv:1801.07699 (2018)
  4. [BPZ84a]
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [BPZ84b]
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34(5–6), 763–774 (1984)ADSMathSciNetCrossRefGoogle Scholar
  6. [BLV16]
    Beneš, C., Lawler, G.F., Viklund, F.: Scaling limit of the loop-erased random walk Green’s function. Probab. Theory Related Fields 166(1), 271–319 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [BSA88]
    Benoit, L., Saint-Aubin, Y.: Degenerate conformal field theories and explicit expressions for some null vectors. Phys. Lett. B215(3), 517–522 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. [CS11]
    Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228(3), 1590–1630 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [CS12]
    Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. [CFL28]
    Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Dub06a]
    Dubédat, J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. [Dub06b]
    Dubédat, J.: Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Rel. Fields 134(3), 453–488 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Dub07]
    Dubédat, J.: Commutation relations for SLE. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Dub15a]
    Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336(2), 695–760 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. [Dub15b]
    Dubédat, J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. 336(2), 761–809 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. [Fel89]
    Felder, G.: BRST approach to minimal models. Nucl. Phys. B 317(1), 215–236 (1989). Erratum ibid. B 324(2):548 (1989)ADSMathSciNetCrossRefGoogle Scholar
  17. [FF90]
    Feĭgin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Representation of Lie Groups and Related Topics, Volume 7 of Advanced Studies in Contemporary Mathematics, pp. 465–554. Gordon and Breach, New York (1990)Google Scholar
  18. [FFK89]
    Felder, G., Fröhlich, J., Keller, G.: Braid matrices and structure constants for minimal conformal models. Commun. Math. Phys. 124(4), 647–664 (1989)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [FP19]
    Flores, S.M., Peltola, E.: Monodromy invariant CFT correlation functions of first column Kac operators. In preparation (2019)Google Scholar
  20. [FK15a]
    Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations, part I. Commun. Math. Phys. 333(1), 389–434 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [FK15b]
    Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations, part II. Commun. Math. Phys. 333(1), 435–481 (2015)ADSzbMATHCrossRefGoogle Scholar
  22. [FK15c]
    Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations, part III. Commun. Math. Phys. 333(2), 597–667 (2015)ADSzbMATHCrossRefGoogle Scholar
  23. [Fom01]
    Fomin, S.: Loop-erased walks and total positivity. Trans. Am. Math. Soc. 353(9), 3363–3583 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [DMS97]
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, Berlin (1997)zbMATHCrossRefGoogle Scholar
  25. [GV85]
    Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58(3), 300–321 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [IK11]
    Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  27. [JJK16]
    Jokela, N., Järvinen, M., Kytölä, K.: SLE boundary visits. Ann. Henri Poincaré 17(6), 1263–1330 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [Kac78]
    Kac, V.: Highest weight representations of infinite dimensional Lie algebras. Proc. ICM Helsinki 1978, 299–304 (1980)MathSciNetzbMATHGoogle Scholar
  29. [KN04]
    Kager, W., Nienhuis, B.: A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115(5), 1149–1229 (2004)ADSzbMATHCrossRefGoogle Scholar
  30. [KM59]
    Karlin, S., McGregor, J.: Coincidence probabilities. Pac. J. Math. 9(4), 1141–1164 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  31. [KKP19]
    Karrila, A., Kytölä, K., Peltola, E.: Conformal blocks, \(q\)-combinatorics, and quantum group symmetry. Annales de l’Institut Henri Poincaré D (2019)Google Scholar
  32. [Kar19]
    Karrila, A.: Multiple SLE type scaling limits: from local to global. Preprint arXiv:1903.10354 (2019)
  33. [Ken00]
    Kenyon, R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185(2), 239–286 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  34. [KW11a]
    Kenyon, R.W., Wilson, D.B.: Boundary partitions in trees and dimers. Trans. Am. Math. Soc. 363(3), 1325–1364 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  35. [KW11b]
    Kenyon, R.W., Wilson, D.B.: Double-dimer pairings and skew Young diagrams. Electr. J. Comb. 18(1), 130–142 (2011)MathSciNetzbMATHGoogle Scholar
  36. [KW15]
    Kenyon, R.W., Wilson, D.B.: Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs. J. Am. Math. Soc. 28(4), 985–1030 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Kim12]
    Kim, J.S.: Proofs of two conjectures of Kenyon and Wilson on Dyck tilings. J. Combin. Theory Ser. A 119(8), 1692–1710 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  38. [KMPW14]
    Kim, J.S., Mészáros, K., Panova, G., Wilson, D.B.: Dyck tilings, increasing trees, descents, and inversions. J. Combin. Theory Ser. A 122(C), 9–27 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  39. [KL07]
    Kozdron, M.J., Lawler, G.F.: The configurational measure on mutually avoiding SLE paths. In: Universality and Renormalization: From Stochastic Evolution to Renormalization of Quantum Fields, Fields Inst. Commun. Amer. Math. Soc., New York (2007)Google Scholar
  40. [KP16]
    Kytölä, K., Peltola, E.: Pure partition functions of multiple SLEs. Commun. Math. Phys. 346(1), 237–292 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. [KP19]
    Kytölä, K., Peltola, E.: Conformally covariant boundary correlation functions with a quantum group. J. Eur. Math. Soc. (2019)Google Scholar
  42. [Law05]
    Lawler, G.F.: Conformally Invariant Processes in the Plane. American Mathematical Society, New York (2005)Google Scholar
  43. [Law91]
    Lawler, G.F.: Intersections of Random Walks. Birkhäuser, Berlin (1991)zbMATHCrossRefGoogle Scholar
  44. [Law14]
    Lawler, G.F.: The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab. 19, 1–13 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  45. [LSW04]
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  46. [LV16]
    Lawler, G.F., Viklund, F.: Convergence of loop-erased random walk in the natural parametrization. Preprint arXiv:1603.05203 (2016)
  47. [LV19]
    Lenells, J., Viklund, F.: Schramm’s formula and the Green’s function for multiple SLE. J. Stat. Phys. 176(4), 873–931 (2019)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [Lin73]
    Lindström, B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5(1), 85–90 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [MS16]
    Miller, J., Sheffield, S.: Imaginary geometry II: reversibility of \(\text{ SLE } _\kappa (\rho _1; \rho _2)\) for \(\kappa \in (0,4)\). Ann. Probab. 44(3), 1647–1722 (2016)MathSciNetCrossRefGoogle Scholar
  50. [PW14]
    Panova, G., Wilson, D.B.: Pfaffian formulas for spanning tree probabilities. Combin. Probab. Comput. 26(1), 118–137 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  51. [Pel19]
    Peltola, E.: Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties. Ann. Inst. H. Poincaré D (2019)Google Scholar
  52. [PW19]
    Peltola, E., Wu, H.: Global and local multiple SLEs for \(\kappa \le 4\) and connection probabilities for level lines of GFF. Commun. Math. Phys. 366(2), 469–536 (2019)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. [Pem91]
    Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19(4), 1559–1574 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  54. [Pon18]
    Poncelet, A.: Schramm’s formula for multiple loop-erased random walks. J. Stat. Mech. Theory Exp. 2018, 103106 (2018)MathSciNetCrossRefGoogle Scholar
  55. [Rib14]
    Ribault, S.: Conformal field theory on the plane. arXiv:1406.4290 (2014)
  56. [RS05]
    Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  57. [Sch00]
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118(1), 221–288 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  58. [SZ10]
    Schramm, O., Zhou, W.: Boundary proximity of SLE. Probab. Theory Relat. Fields 146(3–4), 435–450 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  59. [SW11]
    Sheffield, S., Wilson, D.B.: Schramm’s proof of Watts’ formula. Ann. Probab. 39(5), 1844–1863 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  60. [SZ12]
    Shigechi, K., Zinn-Justin, P.: Path representation of maximal parabolic Kazhdan–Lusztig polynomials. J. Pure Appl. Algebra 216(11), 2533–2548 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  61. [Wil96]
    Wilson, D.: Generating random spanning trees more quickly than the cover time. In: Proceeding of the 28th Annual ACM Symposium on the Theory of Computing, pp. 296–303 (1996)Google Scholar
  62. [Wu18]
    Wu, H.: Hypergeometric SLE: conformal Markov characterization and applications. Preprint arXiv:1703.02022v4 (2018)
  63. [YY11]
    Yadin, A., Yehudayoff, A.: Loop-erased random walk and Poisson kernel on planar graphs. Ann. Probab. 39(4), 1243–1285 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  64. [Zha08]
    Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Alex Karrila
    • 1
  • Kalle Kytölä
    • 1
  • Eveliina Peltola
    • 2
    Email author
  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  2. 2.Section de MathématiquesUniversité de GenèveGeneva 4Switzerland

Personalised recommendations