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- Title
Critical Measures on Higher Genus Riemann Surfaces.
- Authors
Bertola, Marco; Groot, Alan; Kuijlaars, Arno B. J.
- Abstract
Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martínez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green's kernel. We study a max-min problem for the bipolar Green's energy with external fields Re V where dV is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann–Hilbert problem as we outline in the paper.
- Subjects
RIEMANN surfaces; QUADRATIC differentials; RIEMANN-Hilbert problems; ORTHOGONAL polynomials; CLEAN energy; CRITICAL theory
- Publication
Communications in Mathematical Physics, 2023, Vol 404, Issue 1, p51
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-023-04832-4