Energy Landscapes of Multidimensional Random Fourier Series

To any positive number ε and any nonnegative even Schwartz function w : R → R we associate the random function u on them-torus T := R/Z defined as the real part of the random Fourier series ∑ ~ν∈Zm X~ν,ε exp(2πi~ν · ~ θ), whereX~ν,ε are complex independent Gaussian random variables with variancew(ε|~ν|). LetN denote the number of critical points of u. We describe explicitly two constantsC,C′ such that as ε goes to the zero, the expectation of the random variable ε−mNε converges to C, while its variance is extremely small and behaves like C′εm. CONTENTS