## On the eigenvalues of truncations of random unitary matrices.(English)Zbl 1422.60017

Summary: We consider the empirical eigenvalue distribution of an $$m\times m$$ principle submatrix of an $$n\times n$$ random unitary matrix distributed according to Haar measure. Earlier work of D. Petz and J. Réffy [Probab. Theory Relat. Fields 133, No. 2, 175–189 (2005; Zbl 1076.60022)] identified the limiting spectral measure if $$\frac{m} {n}\to \alpha$$, as $$n\to \infty$$; under suitable scaling, the family $$\{\mu _{\alpha }\}_{\alpha \in (0,1)}$$ of limiting measures interpolates between uniform measure on the unit disc (for small $$\alpha )$$ and uniform measure on the unit circle (as $$\alpha \to 1)$$. In this note, we prove an explicit concentration inequality which shows that for fixed $$n$$ and $$m$$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $$\mu _{\alpha }$$ is typically of order $$\sqrt{\frac{\log (m)}{m}}$$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to D. Chafaï et al. [J. Funct. Anal. 275, No. 6, 1447–1483 (2018; Zbl 1407.82045)].

### MSC:

 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects)

### Citations:

Zbl 1407.82045; Zbl 1076.60022
Full Text:

### References:

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