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Boundary Correlations in Planar LERW and UST

  • Alex Karrila
  • Kalle Kytölä
  • Eveliina PeltolaEmail author
Article
  • 42 Downloads

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 \(\kappa =2\).

Notes

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.

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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

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