Search

We consider the double-dimer model in the upper-half plane discretized by the square lattice with mesh size $\delta$. For each point $x$ in the upper half-plane, we consider the random variable $N_\delta(x)$ given by the number of the double-dimer loops surrounding this point. We prove that the normalized fluctuations of $N_\delta(x)$ for a fixed $x$ are asymptotically Gaussian as $\delta\to 0+$. Further, we prove that the double-dimer nesting field $N_\delta(\cdot) - \mathbb{E}\, N_\delta(\cdot)$, viewed as a random distribution in the upper half-plane, converges as $\delta\to 0+$ to the nesting field of CLE(4) constructed by Miller, Watson and Wilson.

We consider the dimer model on a bipartite graph embedded into a locally flat Riemann surface with conical singularities and satisfying certain geometric conditions in the spirit of the work of Chelkak, Laslier and Russkikh, see arXiv:2001.11871. Following the approach developed by Dub\'edat in his work ["Dimers and families of Cauchy-Riemann operators I". In: J. Amer. Math. Soc. 28 (2015), pp. 1063-1167] we establish the convergence of dimer height fluctuations to the compactified free field in the small mesh size limit. This work is inspired by the series of works of Berestycki, Laslier, and Ray (see arXiv:1908.00832 and arXiv:2207.09875), where a similar problem is addressed, and the convergence to a conformally invariant limit is established in the Temperlian setup, but the identification of the limit as the compactified free field is missing. This identification is the main result of our paper.

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the minimal length, such that \[ \max_{y \in M} dist (y, \Sigma) \leq r. \] The inverse problem is to determine whether a given compact connected set $\Sigma$ is a minimizer for some compact $M$ and some positive $r$. Let a Steiner tree $St$ with $n$ terminals be unique for its terminal vertices. The first result of the paper is that $St$ is a minimizer for a set $M$ of $n$ points and a small enough positive $r$. It is known that in the planar case a general Steiner tree (on a finite number of terminals) is unique. It is worth noting that a Steiner tree on $n$ terminal vertices can be not a minimizer for any $n$ point set $M$ starting with $n = 4$; the simplest such example is a Steiner tree for the vertices of a square. It is known that a planar maximal distance minimizer is a finite union of simple curves. The second result is an example of a minimizer with an infinite number of corner points (points with two tangent rays which do not belong to the same line), which means that this minimizer can not be represented as a finite union of smooth curves. Our third result is that every injective $C^{1,1}$-curve $\Sigma$ is a minimizer for a small enough $r>0$ and $M = \overline{B_r(\Sigma)}$. The proof is based on analogues result by Tilli on average distance minimizers. Finally, we generalize Tilli's result from the plane to $d$-dimensional Euclidean space.

For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

We continue the study of the rational Picard group of the moduli space of Hitchin's spectral covers started in P. Zograf's and D. Korotkin's work [11]. In the first part of the paper we expand the ``boundary'', ``Maxwell stratum'' and ``caustic'' divisors introduced in [11] via the set of standard generators of the rational Picard group. This generalizes the result of [11], where the expansion of the full discriminant divisor (which is a linear combination of the classes mentioned above) was obtained. In the second part of the paper we derive a formula that relates two Hodge classes in the rational Picard group of the moduli space of Hitchin's spectral covers.

We prove that the set of $n$-point configurations for which the solution of 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 on which at least two locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially require rely upon the theory of subanalytic sets developed in~\cite{bierstone1988semianalytic}. Motivated by this approach we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replace 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 the dimension $n\dim M-1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to abovementioned results, we study the set of set of $n$-point configurations for which there is a unique solution of the Steiner problem in $\mathbb{R}^d$. We show that this set is path-connected.

Building upon recent results of Dub\'edat (see arXiv:1403.6076) on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on $\Omega^\delta$ as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as $\delta \to 0$, 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 $\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{not} use any RSW-type arguments for double-dimers. The limits $P_L$ of the probabilities $P_L^\delta$ are defined as coefficients of the isomonodormic tau-function studied by Dub\'edat with respect to the Fock--Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability to obtain $L$ from a sample of the nested CLE(4) in $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble., Comment: minor update following referee's comments (42 pages, 5 figures)

Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $\eta$ be an odd theta characteristic on it such that $h^0(C,\eta) = 1$. Pick a point $p$ from the support of $\eta$ and consider the one-dimensional linear system $|\eta + p|$. In general this linear system is base-point free and all its ramification points (i.e. ramification points of the corresponding branched cover $C\to\mathbb P^1\simeq \mathbb PH^0(C,\eta+p)$) are simple. We study the locus in the moduli space of odd spin curves where the linear system $|\eta + p|$ fails to have this general behavior. This locus splits into a union of three divisors: the first divisor corresponds to the case when $|\eta+p|$ has a base point, the second one corresponds to theta characteristics which are not reduced at $p$ (and therefore $|\eta + p|$ must have a triple point at $p$) and the third one corresponds to the case when $|\eta + p|$ has a triple point different from $p$. The second divisor was studied by G. Farkas and A. Verra in\cite{FARo} where its expansion in the rational Picard group was used to prove that the moduli space of odd spin curves is of general type for genus at least $12$. We call the first divisor a Base Point divisor and the third one a Caustic divisor (following Arnold terminology for Hurwitz spases). The objective of this paper is to expand these two divisors via the set of standard generators in the rational Picard group of the moduli space of odd spin curves.

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. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.