Boundary Correlations in Planar LERW and UST

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 \(\kappa =2\).

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Notes

  1. 1.

    See also the update below the conjecture for recent progress.

  2. 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. 3.

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

  4. 4.

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

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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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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.

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).

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.

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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

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