Hexagon titlings with $m\times m$ periodic weightings

Recent studies have investigated the asymptotic behavior of Determinantal Point Processes arising from hexagonal tilings with periodic weightings, employing matrix-valued orthogonal polynomials. In the specific case of triple periodic weightings, A. B. J. Kuijlaars has characterized the corresponding asymptotics using an equilibrium measure for a scalar potential problem with an external field on a genus one Harnack spectral curve (arXiv:2412.03115v1). In this talk, we present a solution to the weak asymptotic problem for matrix-valued orthogonal polynomials associated with tilings of arbitrary periodicity $m$. The proof utilizes the Passare-Leinartas-Tsikh theorem concerning the asymptotic behavior of solutions to multidimensional difference equations. Our solution is expressed in terms of the amoeba of a Harnack spectral curve of genus $g>1$.
This work was supported by the Russian Science Foundation under grant no. 24-11-00196, https://rscf.ru/en/project/24-11-00196/.