论文标题
催化变量方程中的通用单数指数
Universal singular exponents in catalytic variable equations
论文作者
论文摘要
催化方程出现在几种组合应用中，最著名的是在晶格路径的数字和平面图的枚举中。本文的主要目的是表明（所谓的）阳性催化方程的溶液系数的渐近估计值具有通用的渐近行为。特别是，这提供了一个理由，为什么在各种平面地图类中尺寸$ n $的地图数量渐近地增长，就像$ c \ cdot n^{ - 5/2}γ^n $，适用于合适的正常数$ c $和$ c $和$γ$。从本质上讲，我们必须区分线性催化方程（其中超指数生长为$ n^{ - 3/2} $）和非线性催化方程式（如平面图所示，我们具有$ n^{ - 5/2} $）。此外，我们为参数提供了相当一般的中心限制定理，即使它们不是阳性的参数，这些参数也可以编码。
Catalytic equations appear in several combinatorial applications, most notably in the numeration of lattice path 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 $c\cdot n^{-5/2} γ^n$, 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.
