关于卡方随机变量的线性组合的有效计算及其在串计数中的应用
On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua
摘要：
卡方随机变量的线性组合出现在广泛的领域。不幸的是，一个封闭的，分析表达式的pdf是未知的。作为这项工作的第一个结果，一个明确的解析表达式的两个伽玛随机变量的总和的密度推导。在此基础上，提出了一种计算卡方随机变量的线性组合的有效算法。得到了误差界的显式表达式。所提出的技术被证明是计算效率，即只有多项式的增长相比，大多数其他方法的指数增长性的条款的数量。它提供了一个巨大的提高精度和显示只有对数增长所需的精度。此外，它适用于更大数量的术语，并且目前是计算数百个术语分布的唯一方法。作为应用，我们发现具有N个标量场的四维超引力随机矩阵模型的本征值涨落概率的指数依赖性具有渐近形式exp（-0.35N）.
英文摘要：
Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the pdf is not yet known. As a first result of this work, an explicit analytic expression for the density of the sum of two gamma random variables is derived. Then a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4d supergravity with N scalar fields is found to be of the asymptotic form exp(-0.35N).
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