The statistical behavior of eigenvalues of large random matrices (i.e. in the limit when the matrix size tends to infinity) has been thoroughly investigated, for probability densities of the form

C \exp{ - Tr V ( M ) }

where V(x) is a smooth, real valued function of the real variable x, and V(M) is defined on matrices by "the usual procedure".

First goal: provide a background and introduction to the above.

But for probability densities in which the TRACE does not appear linearly, the situation is less understood. A simple example is:

C \exp{ ( Tr ( M2 ) )2 }

(i.e. square the trace).

Second goal: explain the source of the complication.

Third goal: Describe results. (Joint work with Misha Stepanov, Univ. of Arizona)