English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$submatrix of an $n$ by $n$ circular unitary matrix by deleting the last $n-p_n$columns and rows. Jiang and Qi (2017) proved that the maximum absolute value ofthe eigenvalues (known as spectral radius) of the truncated matrix, afterproperly normalized, converges in distribution to the Gumbel distribution if$p_n/n$ is bounded away from $0$ and $1$. In this paper we investigate thelimiting distribution of the spectral radius under one of the following fourconditions: (1). $p_n\to\infty$ and $p_n/n\to 0$ as $n\to\infty$; (2).$(n-p_n)/n\to 0$ and $(n-p_n)/(\log n)^3\to\infty$ as $n\to\infty$; (3).$n-p_n\to\infty$ and $(n-p_n)/\log n\to 0$ as $n\to\infty$ and (4). $n-p_n=k\ge1$ is a fixed integer. We prove that the spectral radius converges indistribution to the Gumbel distribution under the first three conditions and toa reversed Weibull distribution under the fourth condition.

Spectral Radii of Truncated Circular Unitary Matrices