Integrating products of quadratic forms

We prove that if $q_1,\dots ,q_m:{\mathbb{R}}^n⟶\mathbb{R}$ 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 $\left(1+ q_1\right) \cdots \left(1+q_m\right)$withrespecttothestandardGaussianmeasurein${\Bbb R}^n$canbeapproximatedwithinrelativeerror$\epsilon >0$ in quasi-polynomial $n^{O(1)}{m}^{O(\ln m-\ln ϵ)}$ time, provided $|q_k(x)| \leq \gamma |x|^2 /r$forsomeabsoluteconstant$\gamma > 0$and$k=1, \ldots, m$\mathrm{.}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 a possible application to testing the feasibility of systems of real quadratic equations.