Conservation constraints on random matrices

Abstract

We study the random matrices constrained by the summation rules that are present in the Hessian of the potential energy surface in the instantaneous normal mode calculations, as a result of momentum conservation. In this paper, we analyse the properties related to such conservation constraints in two classes of real symmetric matrices: one with purely row-wise summation rules and the other with the constraints on the blocks of each matrix, which underscores partially the spatial dimension. We show explicitly that the constraints are removable by separating the degrees of freedom of the zero-eigenvalue modes. The non-spectral degrees of freedom under the constraints can be realized in terms of the ordinary constraint-free orthogonal symmetry but with the rank deducted by the block dimension. We propose that the ensemble of real symmetric matrices with full randomness, constrained by the summation rules, is equivalent to the Gaussian orthogonal ensemble (GOE) with lowered rank. Independent of the joint probability distribution, the Jacobian contributed by the transformation from the matrix coordinates to the spectral coordinates, possesses the same spectral-variable dependence in these two cases. We verify numerically that the small-separation asymptotic in the nearest-neighbour level spacing distribution is indeed governed by the same symmetry power factor, β = 1, as that of the GOE, under conservation constraints and other underlying spatial correlations.