  We investigate the maximal size of distinguished submatrices of a Gaussian
random matrix. Of interest are submatrices whose entries have average greater
than or equal to a positive constant, and submatrices whose entries are
well-fit by a two-way ANOVA model. We identify size thresholds and associated
(asymptotic) probability bounds for both large-average and ANOVA-fit
submatrices. Results are obtained when the matrix and submatrices of interest
are square, and in rectangular cases when the matrix submatrices of interest
have fixed aspect ratios. In addition, we obtain a strong, interval
concentration result for the size of large average submatrices in the square
case. A simulation study shows good agreement between the observed and
predicted sizes of large average submatrices in matrices of moderate size.
