Analyticity of the Planar Limit of a Matrix Model

Using Chebyshev polynomials combined with some mild combinatorics, we provide an alternative approach to the analytical and formal planar limits of a random matrix model with a 1-cut potential V. For potentials \({V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n}\), 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 ≥ 1 (on the appropriate spaces), the planar limit is also an analytic function in infinitely many variables and we give quantitative versions of where this is defined. Particularly useful in enumerative combinatorics are the gradings of \({V,V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n}\) and \({V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n}\). The associated planar limits F(t) as functions of t count planar diagram sorted by the number of edges respectively faces. We point out a method of computing the asymptotic of the coefficients of F(t) using the combination of the wzb method and the resolution of singularities. This is illustrated in several computations revolving around the important extreme potential \({V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)}\) and its variants. This particular example gives a quantitative and sharp answer to a conjecture of ’t Hooft’s, which states that if the potential is analytic, the planar limit is also analytic.