On the concentration of random multilinear forms and the universality of random block matrices

• 成果类型: Article
• 署名作者: Nguyen, Hoi H.; O'Rourke, Sean
• 署名单位: University System of Ohio; Ohio State University; Yale University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0567-7
• 发表日期: 2015
• 页码: 97-154
• 关键词: littlewood-offord theorems; random symmetric-matrices; circular law; condition number; probability; INVERTIBILITY; singularity; eigenvalues

摘要

The circular law asserts that if is a matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of converges almost surely to the uniform distribution on the unit disk as tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies on an inverse-type result for the concentration of linear operators and multilinear forms.