Skip to main content
SpringerLink
Log in

Boundary Correlations in Planar LERW and UST

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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 \(\kappa =2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46

Similar content being viewed by others

Notes

  1. See also the update below the conjecture for recent progress.

  2. This is the first proof of the existence of positive pure partition functions of multiple \(\mathrm {SLE}\)s, implying in particular the existence of the extremal (local) multiple \(\mathrm {SLE}\)s. After the present work, the existence of positive pure partition functions for \(\kappa \in (0,6]\) has been proven by \(\mathrm {SLE}\) methods [PW19, Wu18], which however do not yield explicit formulas.

  3. The recent works [PW19, Wu18] show the positivity for \(\kappa \le 6\).

  4. For \(\kappa <8\) we have \(\Delta < \Delta '\), whence the terminology leading and subleading for \(\Delta \) and \(\Delta '\), respectively.

References

  1. Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the Conference ‘Conformal Invariance and Random Spatial Processes’, Edinburgh (2003)

  2. Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6), 1125–1163 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Beffara, V., Peltola, E., Wu, H.: On the uniqueness of global multiple SLEs. Preprint. arXiv:1801.07699 (2018)

  4. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  ADS  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benoit, L., Saint-Aubin, Y.: Degenerate conformal field theories and explicit expressions for some null vectors. Phys. Lett. B215(3), 517–522 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228(3), 1590–1630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubédat, J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Dubédat, J.: Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Rel. Fields 134(3), 453–488 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dubédat, J.: Commutation relations for SLE. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336(2), 695–760 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Dubédat, J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. 336(2), 761–809 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Felder, G.: BRST approach to minimal models. Nucl. Phys. B 317(1), 215–236 (1989). Erratum ibid. B 324(2):548 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  17. 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)

  18. Felder, G., Fröhlich, J., Keller, G.: Braid matrices and structure constants for minimal conformal models. Commun. Math. Phys. 124(4), 647–664 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Flores, S.M., Peltola, E.: Monodromy invariant CFT correlation functions of first column Kac operators. In preparation (2019)

  20. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  ADS  MATH  Google Scholar 

  22. 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)

    Article  ADS  MATH  Google Scholar 

  23. Fomin, S.: Loop-erased walks and total positivity. Trans. Am. Math. Soc. 353(9), 3363–3583 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  25. Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58(3), 300–321 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  27. Jokela, N., Järvinen, M., Kytölä, K.: SLE boundary visits. Ann. Henri Poincaré 17(6), 1263–1330 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Kac, V.: Highest weight representations of infinite dimensional Lie algebras. Proc. ICM Helsinki 1978, 299–304 (1980)

    MathSciNet  MATH  Google Scholar 

  29. Kager, W., Nienhuis, B.: A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115(5), 1149–1229 (2004)

    Article  ADS  MATH  Google Scholar 

  30. Karlin, S., McGregor, J.: Coincidence probabilities. Pac. J. Math. 9(4), 1141–1164 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  31. Karrila, A., Kytölä, K., Peltola, E.: Conformal blocks, \(q\)-combinatorics, and quantum group symmetry. Annales de l’Institut Henri Poincaré D (2019)

  32. Karrila, A.: Multiple SLE type scaling limits: from local to global. Preprint arXiv:1903.10354 (2019)

  33. Kenyon, R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185(2), 239–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kenyon, R.W., Wilson, D.B.: Boundary partitions in trees and dimers. Trans. Am. Math. Soc. 363(3), 1325–1364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kenyon, R.W., Wilson, D.B.: Double-dimer pairings and skew Young diagrams. Electr. J. Comb. 18(1), 130–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kim, J.S.: Proofs of two conjectures of Kenyon and Wilson on Dyck tilings. J. Combin. Theory Ser. A 119(8), 1692–1710 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. 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)

  40. Kytölä, K., Peltola, E.: Pure partition functions of multiple SLEs. Commun. Math. Phys. 346(1), 237–292 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Kytölä, K., Peltola, E.: Conformally covariant boundary correlation functions with a quantum group. J. Eur. Math. Soc. (2019)

  42. Lawler, G.F.: Conformally Invariant Processes in the Plane. American Mathematical Society, New York (2005)

  43. Lawler, G.F.: Intersections of Random Walks. Birkhäuser, Berlin (1991)

    Chapter  MATH  Google Scholar 

  44. Lawler, G.F.: The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab. 19, 1–13 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lawler, G.F., Viklund, F.: Convergence of loop-erased random walk in the natural parametrization. Preprint arXiv:1603.05203 (2016)

  47. Lenells, J., Viklund, F.: Schramm’s formula and the Green’s function for multiple SLE. J. Stat. Phys. 176(4), 873–931 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Lindström, B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5(1), 85–90 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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)

    Article  MathSciNet  Google Scholar 

  50. Panova, G., Wilson, D.B.: Pfaffian formulas for spanning tree probabilities. Combin. Probab. Comput. 26(1), 118–137 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. Peltola, E.: Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties. Ann. Inst. H. Poincaré D (2019)

  52. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19(4), 1559–1574 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  54. Poncelet, A.: Schramm’s formula for multiple loop-erased random walks. J. Stat. Mech. Theory Exp. 2018, 103106 (2018)

    Article  MathSciNet  Google Scholar 

  55. Ribault, S.: Conformal field theory on the plane. arXiv:1406.4290 (2014)

  56. Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118(1), 221–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  58. Schramm, O., Zhou, W.: Boundary proximity of SLE. Probab. Theory Relat. Fields 146(3–4), 435–450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sheffield, S., Wilson, D.B.: Schramm’s proof of Watts’ formula. Ann. Probab. 39(5), 1844–1863 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. Shigechi, K., Zinn-Justin, P.: Path representation of maximal parabolic Kazhdan–Lusztig polynomials. J. Pure Appl. Algebra 216(11), 2533–2548 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. 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)

  62. Wu, H.: Hypergeometric SLE: conformal Markov characterization and applications. Preprint arXiv:1703.02022v4 (2018)

  63. Yadin, A., Yehudayoff, A.: Loop-erased random walk and Poisson kernel on planar graphs. Ann. Probab. 39(4), 1243–1285 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Espoo, Finland

    Alex Karrila & Kalle Kytölä

  2. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case Postale 64, 1211, Geneva 4, Switzerland

    Eveliina Peltola

Authors
  1. Alex Karrila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kalle Kytölä
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eveliina Peltola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eveliina Peltola.

Additional information

Communicated by H. Duminil-Copin

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A. Example Boundary Visit Formulas

Appendix A. Example Boundary Visit Formulas

In this appendix, we provide a few examples of the boundary visit amplitudes \(\zeta \) and the order-refined amplitudes \(\zeta _\omega \) with small numbers of marked points.

1.1 A.1. One boundary visit

For one boundary visit, \(N' = 1\), there is only one possible order (so we omit \(\omega \) from the notation). The corresponding amplitude is obtained using replacing Algorithm 5.3:

One can directly check that this function solves the PDE system in Theorem 3.17 and asymptotics properties (5.17).

1.2 A.2. Two boundary visits

For two boundary visits, \(N' = 2\), there are two essentially different cases: boundary visits on the same side or boundary visits on the opposite sides. As representative examples, we consider the orders \(\omega = (-,-)\) and \(\omega = (-,+)\).

For the visits on the same side, using replacing Algorithm 5.3, we find the solution

Up to a coordinate change and an overall multiplicative constant, this formula coincides with the specialization to \(\kappa =2\) of the Schramm–Zhou two-point boundary proximity function of the chordal \(\mathrm {SLE}_{\kappa }\) [SZ10], as well as with an \(\mathrm {SLE}_{\kappa }\) boundary zig-zag amplitude in [JJK16].

For the visits on opposite sides, using replacing Algorithm 5.3, we find the solution

Up to a coordinate change and an overall multiplicative constant, this formula coincides with a specialization to \(\kappa =2\) of an \(\mathrm {SLE}_{\kappa }\) boundary zig-zag amplitude in [JJK16].

Again, one can check that these functions solve the PDE system in Theorem 3.17 and asymptotics properties (5.17) hold.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karrila, A., Kytölä, K. & Peltola, E. Boundary Correlations in Planar LERW and UST. Commun. Math. Phys. 376, 2065–2145 (2020). https://doi.org/10.1007/s00220-019-03615-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03615-0

Search

Navigation