Garoufalidis, Stavros; Popescu, Ionel Analyticity of the planar limit of a matrix model. (English) Zbl 1271.81168 Ann. Henri Poincaré 14, No. 3, 499-565 (2013). Matrix models are integrals of exponentiated potential functions over finite dimensional vector spaces. They come in two flavors: formal and analytic. When the potential is \(V_4=\frac 12 x^2-\frac 14 a_4 x^4\), D. Bessis et al. [Adv. Appl. Math. 1, 109–157 (1980; Zbl 0453.05035)] gave an exact formula for the planar limit of the formal matrix model. Their method was justified by N. M. Ercolani and K. D. T.-R. McLaughlin [Int. Math. Res. Not. 2003, No. 14, 755–820 (2003; Zbl 1140.82307)] using potential theory and the Riemann-Hilbert method.Using Chebyshev polynomials combined with some mild combinatorics, the authors provide an alternative approach to the planar limits of the analytical and formal matrix models with a 1-cut potential. For potentials \(V=\frac 12 x^2-\sum_{n\geq 1}\frac 1n a_n x^n\), as a power series in all \(a_n\), the formal Taylor expansion of the analytic planar limit is exactly the formal planar limit. In the case \(V\) is analytic in infinitely many variables \(\{a_n \}_{n \geq 1}\), the planar limit is also an analytic function in infinitely many variables. These results confirm a conjecture of ’t Hooft, which states that if the potential is analytic, the planar limit is also analytic.In enumerative combinatorics, two gradings of \(V\) are useful: the edge grading \(V_e=\frac 12 x^2-\sum_{n\geq 1}\frac 1n a_n t^{\frac{n}{2}} x^n\) and the face grading \(V_f=\frac 12 x^2-\sum_{n\geq 3}\frac 1n a_n t^{\frac{n}{2}-1} x^n\). The associated planar limits as functions of \(t\) count planar diagram sorted by the number of edges respectively faces. The authors point out a method of computing the asymptotic of the coefficients of the associated planar limits using the combination of the \(wzb\) method and the resolution of singularities, reestablishing some results of P. M. Bleher and A. R. Its [Ann. Inst. Fourier 55, No. 6, 1943–2000 (2005; Zbl 1135.82016)]. Reviewer: Ren Guo (Corvallis) Cited in 1 ReviewCited in 6 Documents MSC: 81T99 Quantum field theory; related classical field theories 05C30 Enumeration in graph theory 15B52 Random matrices (algebraic aspects) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory Keywords:matrix model; planar limit; analytic function; Chebyshev polynomial; ribbon graph enumeration Citations:Zbl 0453.05035; Zbl 1140.82307; Zbl 1135.82016 PDFBibTeX XMLCite \textit{S. Garoufalidis} and \textit{I. Popescu}, Ann. 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