UC Santa Cruz

This paper demonstrates an introduction to the statistical distribution of eigenval-

ues in Random Matrix theory. Using mathematical analysis and probabilistic measure

theory instead of statistical methods, we are able to draw conclusions on large dimen-

sional cases and as our dimensions of the random matrices tend to innity. Applications

of large-dimensional random matrices occur in the study of heavy-nuclei atoms, where

Eigenvalues express some physical measurement or observation at a distinct state of

a quantum-mechanical system. This specically motivates our study of Wigner Ma-

trices. Classical limit theorems from statistics can fail in the large-dimensional case

of a covariance matrix. By using methods from combinatorics and complex analysis,

we are able to draw multiple conclusions on its spectral distributions. The Spectral

distributions that arise allow for boundedness to occur on extreme eigenvalues.