Analytic subordination theory of operator-valued free additive\n convolution and the solution of a general random matrix problem

2013/03/13 by Serban T. Belinschi, Tobias Mai, Belinschi, Serban +3 · 1 citation
Mathematics · #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1303.3196

Abstract

We develop an analytic theory of operator-valued additive free convolution in\nterms of subordination functions. In contrast to earlier investigations our\nfunctions are not just given by power series expansions, but are defined as\nFrechet analytic functions in all of the operator upper half plane.\nFurthermore, we do not have to assume that our state is tracial. Combining this\nnew analytic theory of operator-valued free convolution with Anderson's\nselfadjoint version of the linearization trick we are able to provide a\nsolution to the following general random matrix problem: How can we calculate\nthe asymptotic eigenvalue distribution of a polynomial evaluated in independent\nrandom matrices with known asymptotic eigenvalue distributions?\n

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