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

单位圆盘中的随机级数、随机矩阵理论和高斯乘法混沌

• 批准号: RGPIN-2020-04974
• 负责人: Paquette, Elliot
• 金额: $ 1.89万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740323
• 关键词: Random; series; unit; disk; random

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.

概率可以说是对统计独立性的研究，既研究这种独立性如何在有时曲折的变换后表现出来的现象学，又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。随机级数理论的发展很大程度上是为了理解幂级数或三角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完成的。随机序列的大部分理论都集中在序列的分析属性、有界性或平滑性上。对于这些物体来说，仍然存在重要的几何问题：这些随机系列的图像可以说些什么？是否有一个一致的理论可以回答这个问题，就像有一个关于随机过程有界性的理论一样？该提案旨在针对这个问题制定随机系列中一些悬而未决问题的答案。它希望通过转向分支过程来找到这些答案。可以针对高斯随机序列提出的大多数问题都有一个启发式答案，可以根据具有时变方差分布的高斯分支随机游走来制定。其中一些问题本身就是分支过程中有趣的开放问题，我们建议对这些模型进行一些研究。它还试图将随机序列的新理论与其他几何概率问题联系起来，围绕高斯乘法混沌，在某些情况下可以被视为某个随机序列的指数。这定义了具有有趣的分形属性的随机度量，并且它出现在许多不同的上下文中。它也自然地与曼德尔布罗特的随机级联理论联系在一起，这也许是随机分形最典型的例子。该提案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高斯情况共享属性。是否有更通用的随机分形理论，包括高斯乘法混沌和其他随机分形？最后，该提案旨在扩展随机矩阵特征多项式与高斯乘法混沌的联系。令人惊讶的是，当矩阵的维数趋于无穷大时，随机矩阵的特征多项式与高斯幂级数具有某种神奇的精确联系，并且它们的特征多项式与这些高斯乘法混沌具有精确的联系。尽管它们在许多统计意义上与高斯幂级数相距甚远。