Integrating Products of Quadratic Forms

Abstract

We prove that if $q_1,\dots ,q_m:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ are quadratic forms in variables $x_1,\dots ,x_n$ such that each $q_k$ depends on at most r variables and each $q_k$ has common variables with at most r other forms, then the average value of the product $(1+q_1)\cdots (1+q_m)$ with respect to the standard Gaussian measure in ${\mathbb {R}}^n$ can be approximated within relative error $\epsilon >0$ in quasi-polynomial $n^{O(1)}\hspace{0.55542pt}m^{O(\ln m-\ln \epsilon )}$ time, provided $|q_k(x)|\leqslant \gamma \hspace{0.33325pt}\Vert x\Vert ^2 /r$ for some absolute constant $\gamma > 0$ and $k=1, \ldots , m$. The integral in question is viewed as the independence polynomial of an auxiliary weighted graph and then the method of polynomial interpolation is applied. When $q_k$ are interpreted as pairwise squared distances for configurations of points in Euclidean space, the average can be interpreted as the partition function of systems of particles with mollified logarithmic potentials. We sketch possible applications to testing the feasibility of systems of real quadratic equations and to computing permanents of positive definite Hermitian matrices.