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Limiting spectral distribution of the product of truncated Haar unitary matrices

Kartick Adhikari

Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India

E-mail Address: kartickmath@gmail.com

Corresponding author.

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and

Arup Bose

http://orcid.org/0000-0002-0303-1369

Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India

E-mail Address: bosearu@gmail.com

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https://doi.org/10.1142/S2010326320500021Cited by:1 (Source: Crossref)

Abstract

Let $A_{n}^{(i)}$ $A_{n}^{(i)}$ , $1 \leq i \leq k$ $1 \leq i \leq k$ , be $k$ $k$ probabilistically independent matrices of order $n_{i} \times n_{i + 1}$ $n_{i} \times n_{i + 1}$ (with $n_{1} = n_{k + 1}$ $n_{1} = n_{k + 1}$ ) which are the left-uppermost blocks of $n \times n$ $n \times n$ Haar unitary matrices. Suppose that $\frac{n}{n_{i}} \to α_{i}$ $\frac{n}{n_{i}} \to α_{i}$ as $n \to \infty$ $n \to \infty$ , with $1 < α_{i} < \infty$ $1 < α_{i} < \infty$ . Using free probability and Brown measure techniques, we find the limiting spectral distribution of $A_{n}^{(1)} \dots A_{n}^{(k)}$ $A_{n}^{(1)} \dots A_{n}^{(k)}$ .

Keywords:

AMSC: 46L54

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