Asymptotic statistics of zeroes for the Lame ensemble

The Lame polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P-0 = Delta, P-1,...,PN-1 acting on C-infinity(S-N). These operat ors naturally depend on parameters and thus constitute an ensemble, In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lame polynomials in various thermodynamic, asymptotic regimes. We g ive results both in the mean and pointwise, for an asymptotically full set of values of the parameters.