Abstract

We prove that if q1,... ,qm : Rn ?? R are quadratic forms in variables x1,... ,xn such that each qk depends on at most r variables and each qk has common variables with at most r other forms, then the average value of the product (1 + q1)���(1 + qm) with respect to the standard Gaussian measure in Rn can be approximated within relative error ? > 0 in quasi-polynomial nO(1)mO(lnm?ln?) time, provided |qk(x)| ? ?kxk2/r for some absolute constant ? > 0 and k = 1,... ,m. When qk 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.

Description