Free Probability and Random Matrices

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A fundamental problem in many branches of science and engineering is to describe and analyse the distributions of a correlated set of random points, often called a point process. The simplest example of this problem is to independently sample from the same probability distribution. The statistics of this example can be analysed by the laws of classical probability going back to De Moivre. However in many examples the points are correlated in subtle and complicated way and we still want to be able to describe these correlations.******An important example is in the study of correlation matrices. In these matrices the (i,j) entry is given by the correlation between the i-th and j-th variables. By examining the singular numbers of the matrix one can distinguish signal from noise or the absence of correlations between the variables.******In wireless communication theory we have to analyse networks where there are multiple transmitting and receiving antennas. A goal here is to estimate the capacity of the channel and again it is the singular value or eigenvalue distribution that is used.******In quantum information theory it is important to detect entangled states. To solve this problem some mathematical methods have been found called entanglement detectors that announce that a matrix is entangled. One such detector is the partial transpose, for if the partial transpose of a positive matrix fails to be positive then the matrix was entangled. As positivity is an eigenvalue question we are again led to eigenvalue distributions.******Thus the general problem in random matrix theory is to find the eigenvalue distribution of the matrix. Since the entries of the matrix are random the eigenvalues are random. So each instance of the matrix gives n eigenvalues and we get a random set of n points. We shall consider self-adjoint matrices in which case the eigenvalues are all real and so we get a random probability measure on the real line. With random measure we have random moments and these random moments have correlations, skewness, kurtosis, and all higher cumulants.******Typically these higher cumulants are very complicated and thought to be too difficult to analyse. However it was noticed that as the number of points increases the complexity melts away and magically simple and beautiful combinatorial pictures emerge. These are the non-crossing partitions or planar graphs.******The goal of this proposal is to extend my previous work on analysing the asymptotics of these higher cumulants, both combinatorially and in the context of analytic functions. The training component of the proposed research will provide students with an understanding of free probability and random matrices theory which will enable them to pursue careers in academia as well as mathematical finance, wireless communication, and quantum computation.**