On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua

TL;DR

This paper introduces an efficient computational method for the distribution of linear combinations of chi-square variables, with applications in counting string vacua, providing explicit formulas, error bounds, and superior accuracy for large numbers of terms.

Contribution

It derives an explicit density formula for the sum of two gamma variables and develops a polynomial-growth algorithm for linear combinations of chi-square variables, enabling computations for hundreds of terms.

Findings

The method is computationally efficient with polynomial growth.

It achieves high accuracy with logarithmic growth in precision.

Applied to supergravity models, it reveals exponential eigenvalue fluctuation dependence.

Abstract

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).