This talk presents several new results concerning the asymptotic behavior of
large average submatrices of an $n \times n$ Gaussian random matrix.  We begin
by considering with the average and joint distribution of the (globally
optimal)
$k \times k$ submatrix having largest average value.  We then consider
submatrices with dominant row and column sums, which arise as the local optima
of a useful iterative search procedure for large average submatrices. We
present
an analysis of the number $L_n(k)$ of locally optimal $k \times k$ submatrices,
beginning with the asymptotic behavior of the mean and the variance of
$L_n(k)$
for fixed k and increasing n. The principal result of the talk is a Gaussian
central limit theorem for $L_n(k)$.