矩阵模型平面极限的解析性
Analyticity of the planar limit of a matrix model
摘要：
利用Chebyshev多项式和一些温和的组合数学方法，给出了一个新的具有单切势$V$的随机矩阵模型的平面极限解析公式。对于势函数$V(x)=x^{2}/2-\sum_{n\ge1}a_{n}x^{n}/n$，作为所有$a_{n}$中的幂级数，解析平面极限的形式泰勒展开式就是形式平面极限。在$V$的情况下是解析的无穷多个变量$\{a_{n}\}_{n\ge1}$（在适当的空间），平面极限也是一个解析函数无穷多个变量，我们给定量版本，这是定义。在枚举组合学中特别有用的是$V$、$V_{t}(x)=x^{2}/2-\sum_{n\ge1}a_{n}t^{n/2}x^{n}/n$和$V_{t}(x)=x^{2}/2-\sum_{n\ge3}a_{n}t^{n/2 -1}x^{n}/n$的分级。相关的平面极限$F(t)$作为$t$的函数计数平面图，该平面图分别由边的数目和面的数目排序。本文将\emph{wzb}方法与奇异性分解相结合，给出了计算$F(t)$系数渐近性的一种方法。这在围绕重要的极端电位$V_{t}(x)=x^{2}/2+\log(1-\sqrt{t}x)$及其变体的几个计算中得到了说明。这个特殊的例子给出了一个定量和尖锐的答案猜想的特霍茨其中指出，如果潜在的是分析，平面极限也是分析。
英文摘要：
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.
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