References & Further Reading
References
- R. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications, Cambridge University Press, 2011
The primary reference for this chapter. Covers Marchenko-Pastur, Stieltjes transform, and deterministic equivalents with wireless applications.
- A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications, Now Publishers (Foundations and Trends in Communications and Information Theory), 2004
Excellent tutorial monograph connecting RMT to MIMO capacity.
- Z. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, Springer, 2nd ed., 2010
Rigorous mathematical treatment of the Marchenko-Pastur law and Stieltjes transform.
- V. A. Marchenko and L. A. Pastur, Distribution of Eigenvalues for Some Sets of Random Matrices, 1967
The original paper establishing the law that now bears their name.
- E. Telatar, Capacity of Multi-Antenna Gaussian Channels, 1999
Landmark paper connecting MIMO capacity to random matrix eigenvalue distributions.
- M. L. Mehta, Random Matrices, Academic Press, 3rd ed., 2004
The definitive mathematical reference on random matrices, from the physics tradition.
- W. Hachem, O. Khorunzhiy, P. Loubaton, J. Najim, and L. Pastur, A New Approach for Mutual Information Analysis of Large Dimensional Multi-Antenna Channels, 2008
Establishes deterministic equivalents for the mutual information of correlated MIMO channels.
- S. Wagner, R. Couillet, M. Debbah, and G. Caire, Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels, 2012
CommIT contribution: applies deterministic equivalents to precoder design in massive MIMO.
- J. Wishart, The Generalised Product Moment Distribution in Samples from a Normal Multivariate Population, 1928
Original paper defining the Wishart distribution for sample covariance matrices.
- T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, 3rd ed., 2003
Classical treatment of Wishart matrices from the statistics perspective.
- E. P. Wigner, Characteristic Vectors of Bordered Matrices with Infinite Dimensions, 1955
Wigner's foundational paper establishing the semicircle law for symmetric random matrices.
- G. Caire, Fundamentals of Stochastic Processes: Lecture Notes, TU Berlin, 2024
Course material. Chapter on random matrix theory motivates the topic from the FSP perspective.
Further Reading
These resources extend the material in this chapter to more advanced random matrix topics and deeper applications in wireless communications.
Tracy-Widom distribution and extreme eigenvalue fluctuations
Johnstone, 'On the distribution of the largest eigenvalue in principal components analysis,' Annals of Statistics, 2001
The edge eigenvalues of Wishart matrices fluctuate around the MP boundary on a scale of $n^{-2/3}$, following the Tracy-Widom distribution. This governs the outage capacity.
Free probability and non-commutative tools
Mingo and Speicher, *Free Probability and Random Matrices*, Springer, 2017
Free probability provides an elegant algebraic framework for computing LSDs of sums and products of independent random matrices, generalizing the Stieltjes transform approach.
Massive MIMO system analysis with RMT
Couillet and Debbah (2011), Chapters 7--9
Detailed application of deterministic equivalents to multi-cell MIMO, pilot contamination, and energy efficiency optimization.
Wigner semicircle law and Gaussian ensembles
Mehta (2004), Chapters 2--5
The Wigner semicircle law is the symmetric-matrix analogue of Marchenko-Pastur. Understanding both laws provides the full picture of spectral convergence for random matrices.