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

概率可以说是对统计独立性的研究，既是关于这种独立性如何在有时曲折的变换后表现出来的现象学，也是有时如何在没有明显内置这种结构的数学问题中找到独立性的现象学。随机数列理论的发展在很大程度上是为了了解幂函数级数或三角函数级数中的系数如何影响极限的定性行为。当这些系数是独立的随机变量时，这是通过询问序列的属性来实现的。随机数列的许多理论都集中在它的分析性质、它的有界性或光滑性上。对于这些物体，重要的几何问题仍然存在：关于这些随机序列的图像，我们可以说些什么？有没有一个理论来回答这个问题是连贯的，就像随机过程的有界性理论一样？这项提议试图在这个问题上以随机数列的形式对一些悬而未决的问题作出回答。它希望通过转向分支流程来开发这些答案。对于高斯随机序列可以提出的大多数问题都有一个启发式答案，它可以用具有时变方差分布的高斯分枝随机游动来表示。其中一些问题本身就是分支过程的有趣的开放问题，我们建议对这些模型进行一些研究。它还试图将关于随机序列的新理论与其他几何概率问题联系起来，围绕着高斯乘性混沌，在某些情况下，它可以被视为某个随机序列的指数。这定义了一种具有有趣的分形性的随机度量，它出现在许多不同的上下文中。它还自然而然地与曼德布罗特的随机级联理论联系在一起，这可能是随机分形最典型的例子。这项建议着眼于其他随机序列的指数，以及它们在多大程度上与特定的高斯情况共享性质。有没有更普遍的随机分形理论，包括高斯乘性混沌和其他随机分形？最后，该建议将随机矩阵的特征多项式的联系扩展到高斯乘性混沌。令人惊讶的是，随机矩阵的特征多项式与高斯级数有着某种神奇的精确联系，当矩阵的维度趋于无穷大时，它们的特征多项式与这些高斯乘法混沌具有确切的联系。这是尽管它们在许多统计意义上与高斯幂级数相对较远。

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.