Limiting Spectral Radii of Circular Unitary Matrices Under Light Truncation

Yu Miao, Yongcheng Qi

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Abstract

Consider a truncated circular unitary matrix which is a pn by pn submatrix of an n by n circular unitary matrix after deleting the last n- pn columns and rows. Jiang and Qi (J Theor Probab 30:326–364, 2017) and Gui and Qi (J Math Anal Appl 458:536–554, 2018) study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix. Some limiting distributions for the spectral radius for the truncated circular unitary matrix have been obtained under the following conditions: (1). pn/ n is bounded away from 0 and 1; (2). pn→ ∞ and pn/ n→ 0 as n→ ∞; (3). (n- pn) / n→ 0 and (n- pn) / (log n) 3→ ∞ as n→ ∞; (4). n- pn→ ∞ and (n- pn) / log n→ 0 as n→ ∞; and (5). n- pn= k≥ 1 is a fixed integer. The spectral radius converges in distribution to the Gumbel distribution under the first four conditions and to a reversed Weibull distribution under the fifth condition. Apparently, the conditions above do not cover the case when n- pn is of order between log n and (log n) 3. In this paper, we prove that the spectral radius converges in distribution to the Gumbel distribution as well in this case, as conjectured by Gui and Qi (2018).

Original language English (US) 2145-2165 21 Journal of Theoretical Probability 34 4 https://doi.org/10.1007/s10959-020-01037-6 Published - Dec 2021

Bibliographical note

Funding Information:
The authors would like to thank the referee for his/her constructive suggestions for revision. The research of Yu Miao was supported in part by NSFC (11971154). The research of Yongcheng Qi was supported in part by NSF Grant DMS-1916014.

Keywords

• Circular unitary matrix
• Eigenvalue
• Extreme value
• Limiting distribution