Winter 2022 course notes: Random Matrix Theory on the Classical Compact Groups

Course content: Introduction to probabilistic aspects of orthogonal, unitary, and (compact) symplectic matrices. Connections to determinantal point processes, orthogonal polynomials, enumerative combinatorics, and analytic number theory. A unifying theme is the theorem of Diaconis and Shahshahani that traces distribute like gaussian random variables as the dimension of matrices becomes large.

References used for course notes:

• [Mec] E. Meckes "The Random Matrix Theory of the Classical Compact Groups" (Primary reference)
• [Dia] P. Diaconis "Patterns in the Eigenvalues"
• [HKPV] J. B. Hough, M. Krishnapur, Y. Peres, B. Virag "Zeros of Gaussian Analytic Functions and Determinantal Point Processes"
• [Sta] R. Stanley "Enumerative Combinatorics" vol. II (Cambridge University Press)
• [MeSn] F. Mezzadri, N.C. Snaith (ed.) "Recent Perspectives in Random Matrix Theory and Number Theory"

Scans of lecture notes:

• [Lecture 1]
  • (Background and motivation)
• [Lecture 2]
  • (Definition of the classical compact groups; [Mec, 1.1])
• [Lecture 3]
  • (More on Sp(2N), Haar measure; [Mec 1.1, 1.2], Royden's "Real Analysis" Ch. 22 for further info on Haar measure)
• [Lecture 4]
  • (Construction of Haar measure on O(N) and other classical compact groups; [Mec 1.2])
• [Lecture 5]
  • (Distribution of matrix entries for O(N), Borel's lemma; [Mec, 2.1]. For another proof of Borel's lemma along with a discussion of how to compute in terms of surface area/uniform distribution on S^{N-1}, see pages 4-5 of these notes of Keith Ball
• [Lecture 6]
  • (Distribution of matrix entries for U(N), Sp(2N); [Mec, 2.1]. Papers mentioned: Diaconis-Freedman, D'Aristotile -Diaconis-Newman
• [Lecture 7]
  • (SO(N) and SU(N); [Mec, 1.1, 1.2]. Rough outline of relationship between SO(3) and SU(2), and Spin(N); a source for this is Ch. VII.5-7 of Simon's "Representations of Finite and Compact Groups." [An account of the relationship between SO(3) and SU(2) using quaternions more explicitly is Ch. 3.3.1 of Schwichtenberg's "Physics from Symmetry" 2nd Ed.])
• [Lecture 8]
  • (Classification of compact Lie groups; Simon Ch.VII.5-7 for statements and further references. Haar measure on SU(2) and Euler angles; for Euler angles applied instead to O(N) see [Mec, 1.2]. Statement of the Weyl integration formula; [Mec, 3.1])
• [Lecture 9]
  • (Hilbert-Schmidt norm; many sources e.g. [Mec, 1.1]. Lie algebras and the exponential map; [Mec, 1.3], another good source is Hall's "Lie Groups, Lie Algebras, and Representations" Ch. 2. Haar measure in a neighborhood of the identity.)
• [Lecture 10]
  • (Preliminaries for the Weyl integration formula: dimension of U(N) and factorization into diagonal and conjugate pieces; [Mec, 3.1] presents the material in a different way which may be preferred by some)
• [Lecture 11]
  • (Weyl integration formula for U(N); [Mec, 3.1]. Many books on Lie groups will also have a proof in more general language. The approach we have used roughly follows ideas used in a slightly different context in Tao's "Topics in Random Matrix Theory", Ch. 2.6)
• [Lecture 12]
  • (Possible eigenvalues for SO(2N), SO(2N+1), Sp(2N) and first look at eigenvalue distributions for these groups; [Mec, 3.1])
• [Lecture 13]
  • (Weyl integration formula for SO(2N), SO(2N+1), Sp(2N), and determinantal formulas; [Mec, 3.1, 3.2]. A first look at point processes; [HKPV, 1.2])
• [Lecture 14]
  • (Foundations of point processes, definitions and examples; [HKPV, 1.2])
• [Lecture 15]
  • (First intensity measure and k-th intensity measures for point processes; [HKPV 1.2])
• [Lecture 16]
  • (Existence and uniqueness for intensity measures; [HKPV, 1.2] see also Sec. 1 of this survey of Soshnikov for additional information. Integration formulas for point processes and determinants; [HKPV, Ch. 4].)
• [Lecture 17]
  • (Determinantal projection processes, proof of Gaudin's Lemma, application to U(N); [HKPV, Ch. 4] or [Mec, 3.2].)
• [Lecture 18]
  • (Correlation functions for Sp(2N), SO(2N), SO(2N+1); [Mec 3.2]. Large N limit for U(N), [Mec 4.1].)
• [Lecture 19]
  • (Determinantal point processes with more general kernels, and links to independence; [HKPV, Ch. 4]. Note that HKPV deal with trace class kernels, more general than finite rank kernels -- we will discuss these later.)
• [Lecture 20]
  • (Proof of theorem at end of Lecture 19, Macchi's theorem, and characterization of counts of configurations in DPP by Bernoulli sums; [HKPV, Ch.4].)
• [Lecture 21]
  • (Central limit theorem of Soshnikov for general determinantal point processes [HKPV, Ch. 4]. Central limit theorem of Costin-Lebowitz for eigenvalues of the unitary group, and a covariance theorem for linear statistics; [Mec 4.1] -- though her approach to computing variance is rather different than ours.)
• [Lecture 22]
  • (Finishing the proof of the covariance theorem. A quick look at trace class kernels for DPP; [HKPV, Ch. 4] develops this fully. Construction of the sine-kernel process. We will not need trace class kernels in what follows, but it is nice to know that the sine-kernel process is a well-defined mathematical object.)
• [Lecture 23]
  • (Convergence of point processes; a complete but somewhat abstract reference is Kallenberg's "Random Measures" (3rd ed. 1983). Also, tail bounds for determinantal point processes; [HKPV, Ch. 4]. As with lecture 22, the material in this lecture can be seen as a supplement that is useful to understand but not essential for the rest of the course.)
• [Lecture 24]
  • (A statement of the Diaconis-Shahshahani theorem on traces; see the survey [Dia]. Foundations of symmetric function theory; see e.g. [Sta, Ch. 7].)
• [Lecture 25]
  • (Power sum symmetric polynomials are a basis, a product identity, and the Hall inner product; [Sta, Ch. 7.7 and 7.9].)
• [Lecture 26]
  • (Orthogonal bases of symmetric functions and the product identity; [Sta, Ch. 7.9]. Alternating functions and the classical definition of Schur functions; [Sta, Ch. 7.15]. Note that our approach to this material is much quicker than Stanley's since we need only a few results for our application to random matrices.)
• [Lecture 27]
  • (Cauchy identity, and Schur functions as formal power series. We follow a different approach here than is typical, but statements of the results we have proved can be found in [Sta, Ch. 7]. The approach we take discussed briefly in the appendix of that chapter.)
• [Lecture 28]
  • (Orthogonality of Schur polynomials in eigenvalues, and the theorem of Diaconis-Shahshahani on moments of traces of unitary matrices; this material is covered in [Mec, 3.3] though the proof there is different.)
• [Lecture 29]
  • (Distributional implications of Diaconis-Shahshahani moment result; see their paper. Discussion of applications to linear statistics; see this paper of Diaconis-Evans. Discussion of applications to the Strong Szego Theorem and Toeplitz determinants; see Section 6.3 of Simon's "Orthogonal Polynomials on the Unit Circle, vol. I" or Basor's article in [MeSn].)
• [Lecture 30]
  • (Zeros of the zeta function, the GUE Hypothesis for local statistics, the result of Rudnick-Sarnak and the Hughes-Rudnick reformulation; see Heath-Brown's article in [MeSn] for an introduction to the zeta-function, and Hughes' article in [MeSn] for information about local statistics.)
• [Lecture 31]
  • (The explicit formulas of Riemann-Guinand-Weil; information about these can be found in Heath-Brown and Hughes' article in [MeSn], as well as a proof of the explicit formula in Ch. 5.5 of Iwaniec-Kowalski's "Analytic Number Theory". Iwaniec-Kowalski work with a Mellin transform as opposed to a Fourier transform, but this only requires a change of variable to pass from their formula to ours.)
• [Lecture 32]
  • (Bounding error terms in linear statistics; see Hughes' article in [MeSn] or the paper "Linear statistics for zeros of Riemann's zeta function" of Hughes-Rudnick. Mean values of Dirichlet polynomials; see Ch. 9.2 of Iwaniec-Kowalski for more material than we will cover in class.)
• [Lecture 33]
  • (the Bohr identification of prime exponentials with iid random variables, smoothed averages, and variance of bandlimited linear statistics; see Hughes' article in [MeSn] or the paper of Hughes-Rudnick.)
• [Lecture 34] [Supplement to Lecture 34]
  • (Higher moments of the Dirichlet polynomial considered by Hughes-Rudnick; see Hughes article in [MeSn] or the paper of Hughes-Rudnick. An optional supplementary note above fills in most of the details of the tauberian argument sketched on the last page of Notes 34.)
• [Lecture 35]
  • (Matching the Hughes-Rudnick computation with random matrix theory; see Hughes article in [MeSn] or the paper of Hughes-Rudnick.)