Enumeration of maps with the Dumitriu-Edelman model

2025/12/08 by Buc-d'Alché, Thomas
Mathematics · #60B20 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications

paper · doi:10.48550/arxiv.2512.07753

openalex publication_date 2025/12/08 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

We give an expansion in 1/N and β of the cumulants of power sums of the particles of the β-ensemble. This new expansion is obtained using the tridiagonal model of Dumitriu and Edelman. The coefficients of the expansion are expressed in terms of suitably labelled maps introduced by Bouttier, Fusy, and Guitter. Our expansion is of a different nature than the one obtained by LaCroix in is study of the b-conjecture of Goulden and Jackson, and involves only orientable maps. We are able to relate bijectively the first two orders of our expansion to the one of LaCroix using a novel many-to-one mapping that relates suitably labelled planar maps with two minima and maps on the projective plane.

Citations

• Tridiagonal random matrices, an analytic approach
• Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law
• Tridiagonalized GUE matrices are a matrix model for labeled mobiles
• General Tridiagonal Random Matrix Models, Limiting Distributions and Fluctuations

Cited by

• Tridiagonal random matrices, an analytic approach

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