With Professor Harold Widom of UC Santa Cruz, Craig introduced a new probability distribution function, now known as the Tracy-Widom distribution. This originated in the early nineties, when Tracy and Widom studied the behavior of the largest eigenvalue of random matrices. Their results, published in 1994 and the following years, not only entirely revolutionized Random Matrix Theory, but the new probability distribution they discovered turned out to be of central importance in many areas of pure and applied mathematics. Important applications have been found in a wide range of fields, including statistics, materials science, genetics, operations research, and financial mathematics.
We are all familiar with the widely occurring Gaussian distribution, often referred to as the bell curve. Underlying its universality is the all-important Central Limit Theorem, which says that, under very general circumstances, any properly scaled quantity that is the sum of a large number of independent random terms, will have a Gaussian distribution. This phenomenon is so common that the Gaussian distribution with zero mean and unit variance is often called the normal distribution.
There are, however, many important phenomena that are the combined effect of a large number of contributions, not as a simple sum of independent terms, but terms combined in a more complicated fashion. The mathematical example that was studied by Tracy and Widom is the largest eigenvalue of matrix with random entries, in the limit where the size of the matrix tends to infinity. Each entry contributes in some way to the largest eigenvalue but their individual effects are far from being independent. The largest eigenvalue, properly scaled, does not behave ‘normally’. It fluctuates in a predictable way which is not Gaussian. Its distribution was first determined by Tracy and Widom.
Examples of situations where the Tracy-Widom Law has applications are principle component analysis in statistics, queuing theory (applications in computer networks, production lines, customer service applications and many another areas), statistical analysis of genetic data, analysis of algorithms in computer science, analysis of hedge fund performance, stochastic effects in growth phenomena in materials science and the natural world such as crystal growth in the presence of impurities, evolution of populations, and much more. In addition, the Tracy-Widom Law plays an essential role in important new developments in several areas of pure mathematics: orthogonal polynomials and special functions, complex analysis, integrable systems, discrete mathematics, algebraic geometry, and the theory of stochastic processes.
Two very recent examples of applications are
- The recent beautiful experimental observation of yet another system that exhibits the Tracy-Widom distribution: the fluctuations (roughness) of interfaces in nematic liquid crystals by K. A. Takeuchi and M. Sano (arXiv:1001.5121).
- The very recent solution of a long-standing open problem almost simultaneously by two sets of authors: the proof of superdiffusivity of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation and its relation to the weakly asymmetric simple exclusion process and directed random polymers. Both groups, T. Sasamoto and H. Spohn (arXiv:0908.2096 and arXiv:1002.1873) on the one hand, and G. Amir, I. Corwin, J. Quastel (arXiv:1003.0443), make use of and acknowledge the breakthrough results by Tracy and Widom published in 2009, as well as on their famous 1994 paper. That these two groups of authors, who had been studying the KPZ problem for many years, suddenly came up with the solution at the same time has a simple explanation: they needed the Tracy-Widom results of 2009 before they could proceed.
Applications of mathematics are embedded in all quantitative sciences, whether they be pure or applied, physical or social. Most of that mathematics dates back one or more centuries. The Tracy-Widom distribution is a rare example of new sophisticated mathematics that answers important problems in a broadrange of sciences in less than two decades.
Craig Tracy has also played pivotal role in the development of mathematical research in UC Davis over the past two decades. His term as Department chair (1994-98) coincided with the period when the UC Davis Mathematics Department first appeared as a rising star nationally and internationally. In addition to the Steele prize, he has received prestigious prizes in recognition of this work that include: The George Pólya Prize of the Society of Industrial and Applied Mathematics in 2002 and the Nobert Wiener Prize in Applied Mathematics in 2007. He was elected Fellow of the American Academy of Arts and Sciences in 2006, and in 2008-09 he received two prestigious lectureships: the Aisenstadt Chair (University of Montreal) and the Batsheva Fellowship in Natural Sciences and Mathematics (Israel Academy of Sciences).
Photo courtesy of Wikipedia (un-attributed CC-SA). Phase transition between a nematic (left) and smectic A (right) phases observed between crossed polarizers. The black color corresponds to isotropic medium.