Dimers on Riemann surfaces and compactified free field

TL;DR

This paper proves that dimer height fluctuations on certain Riemann surfaces converge to the compactified free field as the mesh size approaches zero, extending previous work to new geometric settings.

Contribution

It establishes the convergence of dimer height fluctuations to the compactified free field on Riemann surfaces with conical singularities, identifying the limit explicitly.

Findings

Convergence of dimer height fluctuations to the compactified free field.

Extension of convergence results to Riemann surfaces with conical singularities.

Main result is the explicit identification of the limit as the compactified free field.

Abstract

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, Proceedings of the London Mathematical Society 126.5 (2023), pp. 1656-1739]. Following the approach developed by Dub\'edat in his work [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 [Nathana\"el Berestycki, Beno\^it Laslier, and Gourab Ray, Annales de l'Institut Henri Poincar\'e D 12.2 (2024), pp. 363-444.] and [Nathana\"el Berestycki, Beno\^it Laslier, and Gourab Ray, Probability and Mathematical Physics 5.4 (2024), pp. 961-1037], 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.