Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size n in various planar map classes grows asymptotically like $c\cdot n^{-5/2}{\gamma }^n$, for suitable positive constants c and γ. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is $n^{-3/2}$) and non-linear catalytic equations (where we have $n^{-5/2}$ as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.

催化方程出现在几个组合应用中，最显着的是晶格路径的枚举和平面图的枚举。本文的主要目的是表明（所谓的）正催化方程解的系数的渐近估计具有普遍的渐近行为。特别是，这提供了为什么各种平面地图类中大小为n的地图数量渐近增长的基本原理$丙\cdot n^{-5/2}{\gamma }^n$，对于合适的正常数c和γ。本质上，我们必须区分线性催化方程（其中次指数增长是$n^{-3/2}$）和非线性催化方程（其中我们有 $n^{-5/2}$如在平面地图中）。此外，我们为可以由催化函数方程编码的参数提供了一个非常通用的中心极限定理，即使它们不是正的。