Free deterministic equivalents, rectangular random matrix models, and operator-valued free probability theory (Q2884854)

Voiculescu's asymptotic freeness results for random matrix models have provided a conceptual framework, as well as powerful tools, for studying the collective limiting behavior of families of random matrices. Let \(X_N,Y_N,U_N,D_N\) be independent random matrices such that \(X_N\) is a self-adjoint Gaussian matrix, \(Y_N\) is a non-self-adjoint Gaussian matrix, \(U_N\) is a Haar-distributed unitary matrix and \(D_N\) is a deterministic matrix. On the other hand, let \(s, c, u, d\) be \(*\)-free operators in a noncommutative probability space such that: \(s\) is a semicircular random variable, \(c\) is circular, \(u\) is a free Haar unitary. If the distribution of \(D_N\) converges to that of \(d\) as \(N \to \infty\), then in fact the joint distribution of \((X_N,Y_N,U_N,D_N)\) converges to that of \((s,c,u,d)\).NEWLINENEWLINESuppose now that the eigenvalue distribution of \(D_N\) does not necessarily converge in the limit \(N \to \infty\). An idea going back to \textit{V. L. Girko} [Theory of stochastic canonical equations. Vol. 1 and 2. Dordrecht: Kluwer Academic Publishers (2001; Zbl 0996.60002) and (2001; Zbl 0996.60003)], which has gained recent attention in the engineering literature, is to replace the system by a ``deterministic equivalent''. Roughly speaking, the idea is to replace the Cauchy transform of the underlying random matrix model by one satisfying a closed system of equations which are determined from the problem in an ad hoc way.NEWLINENEWLINEIn this article, the authors develop a new approach to this problem. They define the ``free deterministic equivalent'' to be \((s,c,u,d)\) as above, except that the distribution of \(d\) is chosen to agree with that of \(D_N\) for a fixed integer \(N\), instead of the limit as \(N \to \infty\) (which may not exist). For the known examples of deterministic equivalents, it turns out that the Cauchy transform of this system actually solves the associated system of equations for the deterministic equivalent. This approach is more conceptual, and allows one to use the tools of free probability theory. As an application, they use free deterministic equivalents to recover a result from [\textit{R. Couillet}, \textit{J. Hoydis} and \textit{M. Debbah}, ``A deterministic equivalent approach to the performance analysis of isometric random precoded systems'', preprint (2010)]. Note that an example of this phenomenon has already been observed by \textit{P. Neu} and \textit{R. Speicher} [J. Stat. Phys. 80, No. 5--6, 1279--1308 (1995; Zbl 1081.82575)].NEWLINENEWLINEThis approach works equally well in the case of multiple matrices \(X_N^{(1)},\dotsc,X_N^{(i_1)},Y_N^{(1)},\dotsc,Y_N^{(i_2)}\), \(U_N^{(1)},\dotsc,U_N^{(i_3)}\), \(D_N^{(1)},\dotsc,D_N^{(i_4)}\). Moreover, the authors show that rectangular random matrices can also be treated with the framework of operator-valued free probability. In particular, they give a generalization of Voiculescu's result on asymptotic freeness for randomly rotated random matrices to the setting of rectangular matrices. An appendix by \textit{T. Mai} provides technical results on Cauchy transforms which are relevant to controlling the error in approximation of the free deterministic equivalent to the original model.