Search

We prove that the set of |$n$| -point configurations for which the solution to the planar Steiner problem is not unique has the Hausdorff dimension at most |$2n-1$| (as a subset of |$\mathbb{R}^{2n}$|⁠). Moreover, we show that the Hausdorff dimension of the set of |$n$| -point configurations for which at least two locally minimal trees have the same length is also at most |$2n-1$|⁠. The methods we use essentially rely upon the theory of subanalytic sets developed in [ 1 ]. Motivated by this approach, we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replaced by an arbitrary analytic Riemannian manifold |$M$|⁠. In this setup, we argue that the set of configurations possessing two locally-minimal trees of the same length either has dimension equal to |$n \dim M - 1$| or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to the above-mentioned results, we study the set of |$n$| -point configurations for which there is a unique solution to the Steiner problem in |$\mathbb{R}^{d}$|⁠. We show that this set is path-connected. [ABSTRACT FROM AUTHOR]

Building upon recent results of Dubedat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations Ωδ to a simply connected domain Ω ⊂ ℂ we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on Ωδ as δ → 0. More precisely, let λ1,..., λn ∈ Ω and L be a macroscopic lamination on Ω \ {λ1,..., λn}, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities PLδ that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on Ωδ converge to a conformally invariant limit PL as δ → 0, for each L. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom(π1(Ω \ {λ1,..., λn}) → SL2(ℂ)) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits PL of the probabilities PLδ are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock-Goncharov lamination basis on the representation variety. The fact that PL coincides with the probability of obtaining L from a sample of the nested CLE(4) in Ω requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble. [ABSTRACT FROM AUTHOR]

The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. [ABSTRACT FROM AUTHOR]