التفاصيل البيبلوغرافية
| العنوان: |
Critical Measures on Higher Genus Riemann Surfaces. |
| المؤلفون: |
Bertola, Marco, Groot, Alan, Kuijlaars, Arno B. J. |
| المصدر: |
Communications in Mathematical Physics; Nov2023, Vol. 404 Issue 1, p51-95, 45p |
| مصطلحات موضوعية: |
RIEMANN surfaces, QUADRATIC differentials, RIEMANN-Hilbert problems, ORTHOGONAL polynomials, CLEAN energy, CRITICAL theory |
| مستخلص: |
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. [ABSTRACT FROM AUTHOR] |
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