Boolean Methods, Expectations, Resolvents, Free Probability

Big data and artificial intelligence are the great ventures of our century. In many cases data comes in the form of huge tables and capturing their essential contents is an important issue. From a mathematicians point of view one can consider a table of data as a matrix with random entries. Moreover, the internals of neural networks, which are an essential tool for the analysis of big data, involve large matrices as well. For these and other reasons the study of random matrices has become a popular endeavour in recent years. One of the basic characteristics of a matrix is the so-called spectrum, that is, its set of eigenvalues, which encodes essential features of the matrix in compact form. It turns out that for a huge class of large random matrices, when the size grows to infinity the spectrum ceases to be random and concentrates in a deterministic limit. Non-commutative free probability provides a formalism which allows to describe this deterministic limit of random matrices: it turns out that under certain symmetry constraints, large stochastically independent random matrices can be effectively approximated by noncommutative "free independent" variables. Using elementary algebraic operations like addition and multiplication one can build complicated matrices out of independent matrices and ask about their spectrum. In the case of simple addition and multiplication the spectrum of the result is well understood in terms of so-called free convolution, but from the point of view of applications, it is cucial to obtain similar understanding of more complicated transformations of independent matrices. The objective of the present project is a kind of "free integral calculus" which will provide efficient algebraic tools for the computation of the asymptotic spectra of arbitrary noncommutative polynomials in independent random matrices. We will not address the issue of convergence, which is well understood, but rather deal with the relevant operations on the level of non-commutative probability (i.e., after going with the matrix size "to infinity"). Our approach includes the case of non-selfadjoint matrices, for which the problem discussed above is much more complicated. The spectrum of non-selfadjoint matrices is a two dimensional subset of the complex plane rather than the real numbers. It is much less understood than in the self-adjoint case and currently is the subject of intensive studies. The present project will also contribute a new approach to the study of large non-selfadjoint random matrices.