Abstract
Let O be chosen uniformly at random from the group of (N+L)×(N+L) orthogonal matrices. Denote by O˜ the upper-left N×N corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues NR(m) of the product matrix O˜1…O˜m, where the matrices {O˜j}j=1m are independent copies of O˜. When L grows in proportion to N, we prove that E(NR(m))= 2mL πarctanh N N+L+O(1),N→∞. We also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where L is fixed with respect to N, known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that E(NR(m))∼cL,mlog(N) as N→∞ and compute the constant cL,m explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).
Highlights
Introduction and main resultsResearch on products of random matrices started in 1960 with the work of Furstenberg and Kesten [22]
More recently there were several new developments related to the opposite regime, namely when the number of factors m is finite and the matrix dimension N tends to infinity, or is finite [5]. Progress in this direction came with the work of Burda et al [10], who computed the spectral density for a product consisting of independent non-Hermitian matrices whose elements are i.i.d. standard complex normal random variables, the complex Ginibre ensemble, in the limit N → ∞ and for finite m
Akemann and Burda [2] discovered that the determinantal structure of the eigenvalue point process known to hold for a single complex Ginibre matrix continues to hold for products of such matrices
Summary
Research on products of random matrices started in 1960 with the work of Furstenberg and Kesten [22]. More recently there were several new developments related to the opposite regime, namely when the number of factors m is finite and the matrix dimension N tends to infinity, or is finite [5] Progress in this direction came with the work of Burda et al [10], who computed the spectral density for a product consisting of independent non-Hermitian matrices whose elements are i.i.d. standard complex normal random variables, the complex Ginibre ensemble, in the limit N → ∞ and for finite m. If A is the random variable whose density is given by the m = 1 case of (1.5), the symmetrised power AmB gives the density (1.5) for any m ≥ 1, where B is an independent Bernoulli random variable on {−1, 1} This type of result is familiar in the study of free probability [11] which is very effective at computing the complex spectrum of a product matrix like X(m) in terms of the spectra of the individual factors. Appendix A includes a discussion about how the approach of the present paper is adapted to the case of products of real Ginibre matrices
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