Limiting behavior of immanants of certain correlation matrix

Abstract

A correlation matrix is a positive semi-definite Hermitian matrix with all diagonals equal to 1. The minimum of the permanents on singular correlation matrices is conjectured to be given by the matrix Yn, all of whose non-diagonal entries are −1/(n−1). Also, Frenzen–Fischer proved that perYn approaches to e/2 as n→∞. In this paper, we analyze some immanants of Yn, which are the generalizations of the determinant and the permanent, and we generalize these results to some other immanants and conjecture most of those converge to 1.