Limiting Spectral Radii of Circular Unitary Matrices Under Light Truncation

Consider a truncated circular unitary matrix which is a p_n by p_n submatrix of an n by n circular unitary matrix after deleting the last n- p_n 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). p_n/ n is bounded away from 0 and 1; (2). p_n→ ∞ and p_n/ n→ 0 as n→ ∞; (3). (n- p_n) / n→ 0 and (n- p_n) / (log n) ³→ ∞ as n→ ∞; (4). n- p_n→ ∞ and (n- p_n) / log n→ 0 as n→ ∞; and (5). n- p_n= 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- p_n 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).