On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them

Dimers on Riemann surfaces and compactified free field

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