Random series in the unit disk, random matrix theory, and the gaussian multiplicative chaos

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Probability is arguably the study of statistical independence, both the phenomenology of how this independence manifests after sometimes tortuous transformations and sometimes how to find independence in mathematical problems that do not obviously have this structure built in. The theory of random series was developed in large part to understand how the coefficients in a power series or trigonometric series influence the qualitative behavior of the limit. This is done by asking for properties of the series when these coefficients are independent random variables. Much of the theory of random series has focused on analytic properties of the series, its boundedness or its smoothness. Important geometric questions remain for these objects: what can be said about the images of these random series? Is there a theory for answering this question which is coherent, in much the way that there is a theory for the boundedness of random processes? This proposal seeks to develop on this question an answer to some outstanding questions in random series. It looks to develop these answers by turning to branching processes. Most questions which can be posed for Gaussian random series have a heuristic answer that can be formulated in terms of a Gaussian branching random walk with time varying variance profile. Some of these questions are interesting open problems for the branching process in their own right, and we propose some study into these models. It also seeks to tie new theory on random series to other geometric probability questions, surrounding the Gaussian multiplicative chaos, which in some cases can be viewed as the exponential of a certain random series. This defines a random measure with interesting fractal properties, and it appears in many different contexts. It also naturally ties into Mandelbrot's theory of random cascades, perhaps the most canonical example of a random fractal. This proposal looks at the exponential of other random series, and to what extent they share properties with the specific Gaussian case. Is there a more general theory of random fractal that includes the Gaussian multiplicative chaos and other random fractals? Finally, this proposal looks to expand the connection of the characteristic polynomial of random matrices to the Gaussian multiplicative chaos. Surprisingly, characteristic polynomials of random matrices have a somewhat miraculous exact connection to Gaussian power series, and their characteristic polynomials have an exact connection to these Gaussian multiplicative chaoses, when the dimension size of the matrix tends to infinity. This is in spite of their being relatively far from Gaussian power series in many statistical senses.