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Abstract
We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple $\mathrm {SLE}(2)$, i.e., an $\mathrm {SLE}(2)$ process weighted by a suitable partition function. By recent results, this also characterizes the “global” scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with $N$ branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence proof for a boundary-visiting UST branch and a boundary-visiting $\mathrm {SLE}(2)$.
Highlights
Schramm–Loewner evolution (SLE) type curves are conformally invariant random curves [39, 38, 11], known or conjectured to describe the scaling limits of random interfaces in many critical planar lattice models [42, 34, 40, 4, 48, 41, 14, 6, 15, 12]
The proof of our main theorem 2.1 provides an important example of the relation between convergence proofs based on local and global multiple SLEs, as discussed in [20, Section 1]: On the one hand, due to recent characterization results for global multiple SLEs, rather short convergence proofs can nowadays be given for various discrete chordal curve models, when conditioned on the pairing of the boundary points by the curves [46, 2]
The analogue of the main result for a boundary-visiting uniform spanning tree (UST) branch is discussed in Section 6, and its interpretation in terms of boundary-visiting SLEs in Appendix C
Summary
Schramm–Loewner evolution (SLE) type curves are conformally invariant random curves [39, 38, 11], known or conjectured to describe the scaling limits of random interfaces in many critical planar lattice models [42, 34, 40, 4, 48, 41, 14, 6, 15, 12]. The proof of our main theorem 2.1 provides an important example of the relation between convergence proofs based on local and global multiple SLEs, as discussed in [20, Section 1]: On the one hand, due to recent characterization results for global multiple SLEs, rather short convergence proofs can nowadays be given for various discrete chordal curve models, when conditioned on the pairing of the boundary points by the curves [46, 2]. Such proofs require as an input the convergence of the corresponding one-curve model to chordal SLE (see [48] on the UST). The analogue of the main result for a boundary-visiting UST branch is discussed in Section 6, and its (non-rigorous) interpretation in terms of boundary-visiting SLEs in Appendix C
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