Integrating Products of Quadratic Forms

We prove that if q_1,… ,q_m:ℝ^n →ℝ are quadratic forms in variables x_1,… ,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)⋯ (1+q_m) with respect to the standard Gaussian measure in ℝ^n can be approximated within relative error ϵ >0 in quasi-polynomial n^O(1)m^O(ln m-lnϵ ) time, provided |q_k(x)|⩽γ‖ x‖ ^2 /r for some absolute constant γ > 0 and k=1, … , 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.