Two-parameter noncommutative Gaussian processes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-237).The reality of billion-user networks and multi-terabyte data sets brings forth the need for accurate and computationally tractable descriptions of large random structures, such as random matrices or random graphs. The modern mathematical theory of free probability is increasingly giving rise to analysis tools specifically adapted to such large-dimensional regimes and, more generally, non-commutative probability is emerging as an area of interdisciplinary interest. This thesis develops a new non-commutative probabilistic framework that is both a natural generalization of several existing frameworks (viz. free probability, q-deformed probability) and a setting in which to describe a broader class of random matrix limits. From the practical perspective, this new setting is particularly interesting in its ability to characterize the behavior of large random objects that asymptotically retain a certain degree of commutative structure and therefore fall outside the scope of free probability. The type of commutative structure considered is modeled on the two-parameter families of generalized harmonic oscillators found in physics and the presently introduced framework may be viewed as a two-parameter deformation of classical probability. Specifically, we introduce (1) a generalized Non-commutative Central Limit Theorem giving rise to a two-parameter deformation of the classical Gaussian statistics and (2) a two-parameter continuum of non-commutative probability spaces in which to realize these statistics. The framework that emerges has a remarkably rich combinatorial structure and bears upon a number of well-known mathematical objects, such as a quantum deformation of the Airy function, that had not previously played a prominent role in a probabilistic setting. Finally, the present framework paves the way to new types of asymptotic results, by providing more general asymptotic theorems and revealing new layers of structure in previously known results, notably in the "correlated process version" of Wigner's Semicircle Law.by Natasha Blitvić.Ph.D