# Projects

## Mid-term Project

You will be asked to read a paper on a topic of interest to you that involves random matrix theory and present it via some mixture of the following perspectives:

• Write a description of greater clarity than the original publication, or

• Devise an improved solution to the problem under consideration, and write up your improvement (with appropriate discussion of the original solution).

• Implement the result in MATLAB® in order to study its performance in practice when the random matrices are finite. Considerations include choice of random matrix result, design of good tests, interpretation of results, and design and analysis of heuristics for improving performance in practice.

## Semester Project

The semester project can be an extension of the mid-term project if it sustains your interest. Otherwise, you will be asked to come up with some insights into a random matrix problem that is of interest to you.

Ideally, the topic you choose will be motivated by a computational problem you need to solve in your research. If the project is large, it may be tackled by a group. Otherwise, feel free to ask us for suggestions and ideas. Our goal is that you are able to get your hands wet trying to solve a problem while using computational and analytical tools that you’ve learned about.

In the past, a student's whole PhD thesis came out of such an exploration. While that might be the exception and rather than the norm we hope we can instil a sense of adventure in you to tackle an interesting random matrix problem and to, hopefully, help you get some useful results out of it.

## Ideas for Projects

TOPICSRELATED PAPERS
Random Growth ProcessesSimulation 1

Simulation 2

On the Second Eigenvalue and Random Walks in Random d-Regular Graphs - Friedman

Random Vicious Walks and Random Matrices - Baik

Toeplitz Determinants, Random Growth and Determinantal Processes - Johansson
Proofs of Semicircle LawA Simple Approach to Global Regime of the Random Matrix Theory - Pastrur

Some Elementary Results around the Wigner Semicircle Law (PDF) - Khorunzhy
Random Matrices with Complex EigenvaluesNon-Hermition Random Matrices (PDF) - Khoruzhenko

Regular Spacings of Complex Eigenvalues in the One-Dimensional Non-Hermitian Anderson Model - Khoruzhenko
Functions for Random MatricesCorrelation Functions, Cluster Functions and Spacing Distributions for Random Matrices - Tracy and Widom

Shape Fluctuations and Random Matrices - Johansson
Random Matrices and PCAOn the Distribution of the Largest Principal Component (PDF) - Johnstone
Communication TheoryMultiuser Receivers, Random Matrices, and Free Probability (PDF) - Tse

Asymptotic Eigenvalue Moments for Linear Multiuser Detection - Verdu

Statistical Analysis of the Capacity of MIMO Frequency Selective Reyleigh Fading Channels (PDF) - Scaglione

The Statistics of the MIMO Frequency Selective Fading AWGN Channel Capacity - Scaglione

Capacity of Multi-antenna Gaussian Channels - Telatar

Complex Random Matrices and Reyleigh Channel Capacity (PDF) - Ratnarajah and Vaillancourt

Spectral Efficiency of CDMA with Random Spreading - Verdu and Shamai

Upper Bounds on the Bit-Error Rate of Optimum Combining in Wireless Systems - Winters and Salz

Bounds and Approximations for Optimum Combining of Signals in the Presence of Multiple Cochannel Interferers and Thermal Noise - Chiani and Win

Environmental Issues for MIMO Capacity (PDF) - Bliss and Forsythe
Free Probability and Addition of Random MatricesMultiuser Receivers, Random Matrices, and Free Probability (PDF) - Tse

Free Probability Theory and Random Matrices (PS) - Speicher

On the Law of Addition of Random Matrices - Pastur and Vasilchuk

Law of Addition in Random Matrix Theory - Zee
Finite FieldsRandom Matrix Theory over Finite Fields (PDF) - Fulman
Array Signal ProcessingInferring the Eigenvalues of Covariance Matrices from Limited, Noisy Data - Everson and Roberts

Large Dimensional Random Matrix Theory for Signal Detection and Estimation in Array Processing (PDF) - Silverstein and Combettes
Random NetworksEigenvalues of Random Power Law Graphs (PDF) - Chung, Lu, and Vu

The Web as a Graph: Measurements, Models and Methods - Kleinberg, et. al.

The Spectra of Random Graphs with Given Expected Degrees (PDF) - Chung, Lu, and Vu

Spectra and Eigenvectors of Scale-free Networks (PDF) - Goh, Kahng, and Kim

Spectra of Real World Graphs (PDF) - Farkas, et. al.
Random GraphsEigenvalues of Random Power Law Graphs (PDF) - Chung, Lu, and Vu

The Spectra of Random Graphs with Given Expected Degrees (PDF) - Chung, Lu, and Vu

Spectra of Real World Graphs (PDF) - Farkas, et. al.
Computational BiologyTheoretical Limitations of Massively Parallel Biology - Szallasi, Periwal, et. al.

RNA Folding and Large N Matrix Theory - Zee
Financial AnalysisExponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimization (PDF) - Pafka, Potters, and Kondor

Noise Dressing of Financial Correlation Matrices (PDF) - Laloux, Cizeau, Bouchaud, and Potters

Random Matrix Theory and Financial Correlations (PDF) - Laloux, Cizeau, Bouchaud, and Potters

Rational Decisions, Random Matrices and Spin Glasses (PDF) - Galluccio, Bouchaud, and Potters
Random Matrices and the Replica Method

Principal Component Analysis Eigenvalue Spectra from Data with Symmetry Breaking Structure (PDF) - Hoyle and Rattray

Asymptotic Corrections to the Wigner Semicircular Eigenvalue Spectrum of a Large Real Symmetric Random Matrix using the Replica Method - Dhesi and Jones

Random Matrices, Free Probability and the Replica Method (PS) - Mueller