Universality in Random Matrix Theory
Ji Oon Lee, Korea Advanced Institute of Science and Technology
Random matrix theory is the study of matrices whose entries are random. It provides a very simple yet powerful tool to analyze strongly correlated systems. Since the pioneering works by Wigner, many remarkable properties of large random matrices have been proved, including universality results that the local eigenvalue statistics of a large class of random matrices show universal pattern independent of the distribution of matrix entries. This is reminiscent of the celebrated central limit theorem, which asserts that the sum of independent random variables exhibits a universal pattern, the Gaussian distribution.
Random matrix theory opens up the new way of understanding universal behaviors in many fields of study. We can find the same statistics, obtained from random matrix theory, from chaotic quantum systems in physics, from principal component analysis in statistics and machine learning, and from wireless communication systems in electrical engineering. Even in number theory, we can observe the same pattern in the distribution of zeroes of Riemann zeta function. The dictionary of such examples of becoming larger and larger, and we can apply the results of random matrix theory to investigate them in a unified manner.