@article{Multi-Z-Shaped:5728,
      recid = {5728},
      author = {Cai, Wenjun},
      title = {On the Spectrum of Singular Values of Multi-Z-Shaped Graph  Matrices},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2023-03},
      pages = {229},
      abstract = {This thesis studies graph matrices. Graph matrices are a  type of random matrices that were invented as a powerful  tool for analyzing large and complicated moment matrices  which often arise in the analysis of the Sum of Square  Hierarchy. They are also useful for other methods involving  higher moments. Formally, a graph matrix is defined as a  function mapping an n×n random input matrix to an output  matrix which is generally larger and more complicated. This  function is described by a small underlying shape. Previous  studies on graph matrices mainly focused on their norm  bounds. In this thesis, we further investigate their  spectrum.

We start with determining the spectrum of  singular values of Z-shaped and multi-Z- shaped graph  matrices when the entries of the random input matrix have  distribution ±1 and the dimension parameter n → ∞. This  result can be seen as an analog of Wigner’s Semicircle Law  in the special case of ±1 distribution, instead of any  arbitrary distribution with mean 0 and variance 1.

We then  generalize our result to multi-Z-shaped graph matrices with  arbitrary input distributions with variance 1 and 0 odd  moments. We achieve this using the ◦R operation, which  mixes two distributions Ω and Ω′ via a random orthogonal  matrix.

This ◦R operation is closely connected to free  probability theory, more precisely the free multiplicative  convolution in the random matrix setting. As a part of our  analysis, we prove a new direct formula for the moments of  the product of two freely independent variables. We also  prove some new results on non-crossing partitions which is  an essential part of free probability theory.},
      url = {http://knowledge.uchicago.edu/record/5728},
      doi = {https://doi.org/10.6082/uchicago.5728},
}