2017/12/26 by Chuan-Tsung Chan, Chan, Chuan-Tsung, Hsiao-Fan Liu +1
Computer Science · Mathematics · #05A15 #15B52 #Advanced Combinatorial Mathematics #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Random Matrices and Applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1712.09231
openalex publication_date 2017/12/26 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28
We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with the quartic matrix model satisfy the discrete type I Painlevé equation. Through the use of generating function techniques, we show that the planar contribution to the free energy is controlled by the Catalan numbers. We also develop a new systematic scheme of calculating higher-genus contributions to the topological expansion of the free energy of matrix models. It is important that our exact solutions are valid for finite-N matrix models and no continuous limits are taken within our approach. To show the advantages of our approach, we provide new results of the topological expansion of the free energy for the finite-N cubic matrix model.