  For a sufficiently nice 2 dimensional shape, we define its approximating
matrix (or patterned matrix) as a random matrix with iid entries arranged
according to a given pattern. For large approximating matrices, we observe that
the eigenvalues roughly follow an underlying distribution. This phenomenon is
similar to the classical observation on Wigner matrices. We prove that the
moments of such matrices converge asymptotically as the size increases and
equals to the integral of a combinatorially-defined function which counts
certain paths on a finite grid. We also consider the case of several
independent patterned matrices. Under a specific set of conditions, these
matrices admit asymptotic freeness with respect to full-filled independent
square random matrices. In our conclusion, we present several open problems.

Probability;Functional Analysis