435 research outputs found
On uniqueness in Steiner problem
We prove that the set of -point configurations for which solution of the
planar Steiner problem is not unique has Hausdorff dimension is at most .
Moreover, we show that the Hausdorff dimension of -points configurations on
which some locally minimal trees have the same length is also at most .
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
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