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Random Matrix Theory: Invariant Ensembles and Universality

Percy Deift and Dimitri Gioev
American Mathematical Society/Courant Institute of Mathematical Sciences
Publication Date: 
Number of Pages: 
Courant Lecture Notes 18
[Reviewed by
Darren Glass
, on

Choose a matrix at random that has certain properties and consider what the largest eigenvalue of such a matrix is. At first glance it is not clear how to make this question precise, let alone why anyone would ever ask such a question. But the theory of random matrices has continued to flourish in recent years due in part to the large number of applications in areas ranging from analytic number theory to nuclear physics. This new book by Percy Deift and Dimitri Gioev is based on graduate courses given at the Courant Institute and the University of Rochester, and considers the mathematical underpinnings of the subject. In particular, the authors introduce results in functional analysis and use these to compute various eigenvalue statistics for unitary, orthogonal, and symplectic ensembles of matrices. They go on to prove the universality of many ensembles, giving a detailed and thorough exposition of many of the results in their research papers. There are also new results in the book giving explicit versions of error estimates.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose research interests include Galois theory, number theory, and cryptography. He can be reached at

Invariant random matrix ensembles: unified derivation of eigenvalue cluster and correlation functions 

  • Introduction and examples
  • Three classes of invariant ensembles
  • Auxiliary facts from functional analysis, Pfaffians, and three integral identities
  • Eigenvalue statistics for the three types of ensembles

Universality for orthogonal and symplectic ensembles 

  • Widom's formulae for the $\beta=1$ and $4$ correlation kernels
  • Large $N$ eigenvalue statistics for the $\beta=1,4$ ensembles with monomial potentials: universality
  • Bibliography
  • Index