Analyticity of the planar limit of a matrix model

TL;DR

This paper develops a new formula for the analytic planar limit of a matrix model with a one-cut potential, demonstrating its analyticity in infinitely many variables and applying it to combinatorial enumeration problems.

Contribution

It introduces a Chebyshev polynomial-based method to explicitly compute the planar limit and proves its analyticity for a broad class of potentials, including those with infinitely many variables.

Findings

Derived a new explicit formula for the planar limit using Chebyshev polynomials.

Proved the analyticity of the planar limit in infinitely many variables for certain potentials.

Applied the method to specific potentials, confirming conjectures about their analytic properties.

Abstract

Using Chebyshev polynomials combined with some mild combinatorics, we provide a new formula for the analytical planar limit of a random matrix model with a one-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 $\{a_{n}\}_{n\ge1}$ (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 \emph{wzb} method and the resolution of singularies. 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 quantitive and sharp answer to a conjecture of t'Hoofts which states that if the potential is analytic, the planar limit is also analytic.