Schramm's formula for multiple loop-erased random walks

Abstract

We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Moebius strip. Schramm's formula is then extended to multiple loop-erased random walks.

59 pages, 19 figures. v2: reformulation of Section 2.3, minor corrections

References in corpus (3)

• Twist operator correlation functions in O(n) loop models
• The scaling limit of two cluster boundaries in critical lattice models
• A generalisation of Schramm's formula for SLE(2)

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