Last updated Jan 26, 2023

Topics in random matrix theory.
Terence Tao

Publication Year: 2012

ISBN-10: 0-8218-7430-6

ISBN-13: 978-0-8218-7430-1

Graduate Studies in Mathematics, vol. 132

American Mathematical Society

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, and “An introduction to measure theory“.

A draft version of the MS can be found here (last updated, Aug 23, 2011; note that the page numbering there differs from that of the published version).  It is based primarily on these lecture notes.

Pre-errata (errors in the draft that were corrected in the published version):

  • p. 20: In Exercise 1.1.11(i), “if and only for” should be “if and only if for”.
  • p. 21: In Exercise 1.1.18, in the definition of convexity, \geq should be \leq.
  • p. 46: In Exercise 1.3.16, Weilandt should be Wielandt. Similarly on p. 47 after Exercise 1.3.9, in Exercise 1.3.22(v) on page 53, on page 137 before (2.82), on page 184 after (2.129), and on page 208 before 2.6.6.  Also, before (1.66), the supremum should be over 1 \leq i \leq n rather than 1 \leq i \leq p.
  • p. 72: All occurrences of 2t/\pi on this page should be \pi t/2.
  • p. 183: The formula (2.127) should be attributed to Dyson ( The three fold way, J. Math. Phys. vol. 3 (1962) pgs. 1199-1215) rather than to Ginibre.  Similarly on pages 251, 259, and 265.
  • p. 225-226: U should be U_0 (several occurrences).  Also, \frac{1}{\sqrt{n}U} should be \frac{1}{\sqrt{n}}U_0 and \frac{1}{\sqrt{n} U_\varepsilon} should be \frac{1}{\sqrt{n}} U_\varepsilon.
  • p. 225, Section 2.8.2: right parenthesis should be added after “sufficient decay at infinity.”
  • p. 228, just before (2.179): “g_n” should be “f_n”
  • p. 231: “lets ignore” should be “lets us  ignore”
  • p. 258: In the second paragraph, d \times d should be n \times n, and X_n should be X_N.
  • Page 6: In parts (iii), (iv) of Lemma 1.1.3, “the \bigwedge_\alpha E_\alpha” should be “then \bigwedge_\alpha E_\alpha“.
  • Page 16: In Example 1.1.9, the Poisson distribution should be classified as having sub-exponential tail rather than being sub-Gaussian.
  • Page 17: In Exercise 1.1.5, \exp(C k^C) should be k^{Ck}.
  • Page 18: Exercise 1.1.10 is incorrect as stated.  Replace the second sentence with “Show that there exists a random variable Y (on the sample space R) taking values in R, such that X is equal almost surely to an extension of Y (to the sample space \Omega), and X' is equal almost surely to an extension of Y (to the sample space \Omega').
  • Page 22: In Exercise 1.1.18(ii), the requirement that the X_i take values in [0,+\infty] should be dropped.
  • Page 29, In Definition 1.1.23, y should lie in R' rather than R.
  • Page 32: In Definition 1.1.29, Remark 1.1.30, and Exercise 1.1.25, “\sigma-compact” should be “\sigma-compact and locally compact”.
  • Page 37: After (1.49), “uniform lower bound” should be “uniform upper bound”, and after the second line of the following display, |x| > \sqrt{n} should be x > \sqrt{n}.
  • Page 41:  In the proof of Theorem 1.3.1, \lambda v^* v should strictly speaking be \lambda(v^*v-1) (though it makes no difference to the remainder of the argument).
  • Page 41: In Exercise 1.3.1, u_j(A)^* u_j(A) should be u_j(A) u_j(A)^*.
  • Page 45: “But, from (1.57)” should be “But, from (1.58)”.
  • Page 49:  After (1.72), \dot u_i^*u_i=0 should be \hbox{Re} \dot u_i^* u_i, and “orthogonal” should be “orthogonal (using the real part of the inner product)”.
  • Page 51: In Section 1.3.6, the role of rows and columns should be reversed in “at least as many rows as columns”.
  • Page 53: In Exercise 1.3.22 (vii), (viii), the eigenvalues \lambda_i should be replaced by singular values \sigma_i.
  • Page 61: In the proof of Theorem 2.1.3, X should be X_i throughout (three occurrences), and {\bf E} \exp( t S_n) = \exp( O( t^2 \sigma^2 ) ) should be {\bf E} \exp( t S_n) \leq \exp( O( t^2 \sigma^2 ) ).  Also, the reference to (2.9) here may be replaced by (2.10).  In the second last line of the proof of Lemma 2.1.2, a closing parenthesis is missing.
  • Page 68: In the last display of Proposition 2.1.9, 2^{-m-1} should be \frac{1}{100(m+1)^2}.
  • Page 69: In Theorem 2.1.10, x_j \in X_j should be x_j \in R_j.  In (2.15), \sigma^2 should be \sigma.
  • Page 70: The footnote “Note that we must have K \leq 1…” should read “Note that we should take K \geq 1 if we wish to allow the variance to actually be able to attain the value 1“.
  • Page 74: In the proof of Lemma 2.1.14, X-A should be A-X (two times).
  • Page 74: In the proof of Lemma 2.1.15, U_B(X') \times \{1\} should be (U_B(X') \backslash U_{A_{X_n}}(X') \times \{1\}, and (\lambda t + (1-\lambda)u, 1-\lambda) should be “one of (\lambda t + (1-\lambda)u, 1-\lambda) or (\lambda t + (1-\lambda)u, 0)“.
  • Page 76: In the proof of Lemma 2.1.16, after (2.24), the expectation in the next two expressions should instead be conditional expectation with respect to X_n.
  • Page 78: In the proof of Proposition 2.1.19, the definitions of X_{i,0} and X_{i,m} are missing absolute value signs (they should be X_{i,0} := X_i {\bf I}(|X_i| \leq 1) and X_{i,m} := X_i {\bf I}(2^{m-1} < |X_i| \leq 2^m ) respectively).  Also, 2^{-m-1} A should  be \frac{A}{100(m+1)^2}.
  • Page 81: In Remark 2.2.2, “central limit” should be “central limit theorem”
  • Page ???: In Lemma 2.3.1, \exp(-CAn) should be -\exp(CA^2 n).
  • In Section 2.4: all references to the “circular law” should be to the “semi-circular law”.
  • Page 88: At and just before (2.4.1), 1+|t| should be |t|.  (Also to avoid having to deal with distributions, one should temporarily truncate 1_{(-\infty,a]} to, say, 1_{[-M,a]} and then let M go to infinity at the end of the argument.)  In the display after (2.40), \hat \eta(t/\varepsilon) should be \hat \eta(\varepsilon t).  Just before the last display in the proof of Theorem 2.2.8, |t| \leq c \varepsilon should be |t| \leq c/\varepsilon.
  • Page 90: In Theorem 2.2.9(i), k should range in 1,2,3,… rather than 0,1,2,… .
  • Page  95: In the first display of the proof of Theorem 2.2.11, o(1) should be O( \frac{1}{\sqrt{n}} {\bf E} |X|^3 ) \sup_{x \in R} |\phi'''(x)|.
  • Page 97:  Near the end of Section 2.2.5: [TaVuKr2010] should be [TaVu2009b].
  • Page 98: In the proof of Theorem 2.2.13, \phi should be assumed to be Lipschitz and not just continuous.
  • Page 99: In the final display, every term should have an expectation symbol {\bf E} attached to it.
  • Page 107: After (2.58), {\bf C}^d should be {\bf C}^n (two occurrences).
  • Page 113: Before (2.62), the \bigvee_{1 \leq i < j \leq n} symbol should be \bigwedge_{1 \leq i < j \leq n}.
  • Page 114: In Proposition 2.3.10, M \|M\|_{op} should be {\bf M} \|M\|_{op} (two occurrences).
  • Page 117: In item (ii) i the list after (2.69), the condition i_1 \neq i_2 should be added.
  • Page 126, second line: the o(1) error term needs to be improved to O(k^2/n).
  • Page 127: In the proof of Lemma 2.3.22, the first arrival can be either a fresh leg or a high multiplicity edge, not simply a fresh leg as stated in the text.  However, this does not affect the rest of the argument.
  • Page 128: For each non-innovative leg, one also needs to record a leg that had already departed from the vertex that one is revisiting; this increases the total combinatorial cost here from k^m to k^{2m} (and the first display should be similarly adjusted).  However, the rest of the argument remains unchanged.  In the last display and the first display of the next page, \max(j+1,k/2) should be \min(j+1,k/2).
  • Page 130: The statement “(2.76) holds” should read “(2.76) fails”.
  • Page ???: In the paragraph before Remark 2.4.1, it should be stated that \mathrm{Pr}({\bf R}) is equipped with the vague topology, and can be viewed as a measurable subset of the compact space \mathrm{Pr}({\bf R} \cup \{\infty\}) of probability measures on the compactification {\bf R} \cup \{\infty\}.
  • Page ???: In the definition of the Stieltjes transform before (2.90), x-z should be z-x.
  • Page 157: In the discussion of classical independence in Section 2.5, “all of {\bf E} f(X), {\bf E} g(Y) vanishes” should be “{\bf E} f(X), {\bf E} g(Y) both vanish”.
  • Page 170: Before Exercise 2.5.10, the constraint X \in L^\infty(\tau) should be X \in {\mathcal A} \cap L^\infty(\tau).
  • Page 174: In Exercise 2.5.15, the additional hypothesis that X and Y are self-adjoint should be added.  In Exercise 2.5.16, add “Show more generally that P(U_1,U_1^*) and Q(U_2,U_2^*) are freely independent for any polynomials P,Q.
  • Page 175: In the second display, an extra right parenthesis should be added to the left-hand side.
  • Page 176: In the proof of Lemma 2.5.20, X_i \in {\mathcal A}_i should be X_i \in {\mathcal A}_i^0.  Also, \tau(X_i Y_{i_1}) should be \tau_i(X_i Y_{i_1}).
  • Page 181: The formulae for \tau(X^k) in Exercises 2.5.20 and Exercises 2.5.21 should be swapped with each other.  Also, the formula for the third cumulant \kappa_3(X) is incorrect; this quantity is in fact equal to the third free cumulant C_3(X) (but \kappa_4(X) and C_4(X) are not equal in general).
  • Page 183: In (2.127), the factor 1! \ldots (n-1)! is missing from the denominator.
  • Page 184: In the paragraph before (2.128), “eigenvalues of M” should be “eigenvalues of M_n“.
  • Page 187-188: The derivation of the Ginibre formula requires modification, because the claim that the space of upper triangular matrices is preserved with respect to conjugation by permutation matrices is incorrect.  Instead, the given data G needs to be replaced by a pair consisting of the random matrix G, together with a random enumeration \lambda_1,\ldots,\lambda_n of the eigenvalues of G, and the factorisation G = U T U^{-1} is then subjected to the constraint that T has diagonal entries \lambda_1,\ldots,\lambda_n in that order.  (To put it another way, one works in an n!-fold cover of the space of matrices with simple spectrum.)  One then performs the analysis in the text, with the enumeration of the eigenvalues of a perturbation of T_0 understood to be the one associated with the diagonal entries of T_0.  (Details may be found at the associated blog entry for this section.)
  • Page 189: In the second paragraph, \varepsilon^{n^2-n} should be (1+o(1)) \varepsilon^{n^2-n}.
  • Page 191: In the last line in the paragraph after (2.137), \frac{1}{|\lambda_i-\lambda_j} should be \log \frac{1}{|\lambda_i - \lambda_j|}.
  • Page 192: In Footnote 52 to Section 2.6.3, the exponent 2 should be 1/2 instead.
  • Page 199: In (2.161), (-1)^{(n+1)/2} should be (-1)^{(n-1)/2}.
  • Page 200: In the definition of \tilde \phi_m, the first factor of \sqrt{n} should be n^{1/4}. In the eigenfunction equation for \tilde \phi_m, L_{1/\sqrt{n}} should be L_{1/n}.
  • Page 201: In (2.162), \frac{1}{n} should be \frac{1}{n^{1/4}}.
  • Page 203: In Exercise 2.6.6, a factor of n^{-1/2} is missing in the O() error term, and |a|^2(x) + |b|^2(x) should be 4 a(0) b(0).   In the penultimate display, n^{(n+1)/2} should be n^{\frac{n}{2} + \frac{1}{4}}.
  • Page 206: In Remark 2.6.8, the n^{1/6} denominator in the first display should instead be in the numerator, and similarly for (2.169); the n^{-1/6} denominator two displays afterwards should similarly be n^{1/6}.
  • Page 212: For the application of Markov’s inequality and through to the next page, all appearances of 8 should be replaced by 8/\delta, and “for at least n/2 values of j” should be “for at least (1-\delta/2)n values of j.  Any appearance of \mathbf{C}^p should instead be \mathbf{R}^p.
  • Page 213: In Exercise 2.7.1, r/\|x\|^2 should be r/\|x\|, the condition \sum_{j: |x_j| \leq \varepsilon^{100}} |x_j|^2 \geq \varepsilon^{10} should be  \sum_{j: |x_j| \leq \varepsilon} |x_j|^2 \geq \eta, the final bound should be \ll_{\eta,\delta} \varepsilon rather than \ll \varepsilon, and |x_j| > 1/2 should be |x_j| > \varepsilon.  The definition of incompressibility should be  \sum_{j: |x_j| \leq \varepsilon} |x_j|^2 \geq \eta, with \eta>0 to be chosen later, in the next display O(\varepsilon)^n should be (O_{\eta,\delta}(\varepsilon))^{(1-\delta/2)n}, and “within \varepsilon\varepsilon^{-200} positions” on the next paragraph should be “within \eta\varepsilon^{-2} positions”.  Finally, in footnote 58, the summation should go up to n rather than to 3 in both occurrences.
  • Page 214: n^{O_\varepsilon(1)} should be n^{O_{\varepsilon,\eta}(1)} (two occurrences), and 2C \varepsilon \sqrt{n} should be 2C \eta \sqrt{n} in Exercise 2.7.2.
  • Page 215: In the last line “Proposition 2.7.3” should be “Proposition 2.7.3 and (2.172)”, and on the next page, O(\sqrt{k})^{-(n-k)} should be \hbox{min}( 1/2, O(k^{-1/2}) )^{n-k} (two occurrences).
  • Page 217: In Exercise 2.7.3, \hbox{dist}(X_i,V_i)^2 should be \hbox{dist}(X_i,V_i)^{-2}.  “\sum_{i=1}^n \sigma_i(M)^{-2} = O(1)” should be “\sum_{i=1}^n \sigma_i(M)^{-2} is comparable to 1.”
  • Page 218.  After the first complete sentence, add “This of course contains the event that \|Mx\| \leq \lambda/\sqrt{n}.”
  • Page 220: In Section 2.7.5, all occurrences of \sqrt{n} \sigma_n(M) should be n \sigma_n(M)^2.
  • Page 223, in Theorem 2.8.1: add “in the vague topology” after “converges”.
  • Page 225: In Section 2.8.2, all integrals should be over {\bf C} rather than {\bf R}.  In Exercise 2.8.3, it should be noted that \frac{1}{w-z} is interpreted in a principal value sense.
  • Pages 226-227: All occurrences of \frac{1}{\sqrt{n} M_n} should be \frac{1}{\sqrt{n}} M_n.
  • Page 228: All occurrences of “operator norm” should be “spectral radius”.
  • Page ???: In Theorem 2.8.3, add the hypotheses that \|\frac{1}{\sqrt{n}} M_n\|_{op} = O(1) almost surely, and that \mu is compactl supported.
  • Page 237: In Proposition 3.1.5, “same distribution as \mu” should be “same distribution as (X_t)_{t \in \Sigma}.  Similarly in Proposition 3.1.16.
  • Page 246 The statement and proof of Theorem 3.1.16 have a number of issues.  A corrected version can be found at this blog post.
  • Page 251: In Exercise 3.1.11, t^{-n^2/2} should be t^{-n/2}.  In (3.12) and the preceding display, n! should be (n-1)!.
  • Page ???: In the last part of Section 2, the threshold \lambda for X_1 + \dots + X_n should be \lambda \sigma, and the factors of e^{-t\lambda} should similarly be e^{-t\lambda\sigma}.
  • Page 267: In the display after “Hermite polynomial”, \frac{d}{dx^2} should be \frac{d^n}{dx^n}.
  • Page 269: In Section 3.4.2, the expansion into Haar coefficients of \sum_j \frac{x_j^2}{2} should instead be of \sum_j x_j^2.
Thanks to Sviatoslav Archava, Giovanni Barbarino, Rex Cheung, Nicholas Christofferson, Nick Cook, Jesus A Dominguez, Peter Forrester, Yihan Gao, Stephen Ge, Paul Jung, Gautam Kamath, keej, Miklós Kornyik, Fan Lau,  Euiwoong Lee, Yijia Liu, John Lentfer, Travis Martin, Ramis Movassagh, Doron Puder, Al Ray, Ilya Razenshteyn, Adam Sawicki, Yeonjong Shin, Frieder Simon, Weiji Su, Ambuj Tewari, Dominik Tomecki, tornado92, Danny Wood, and Felix V. for corrections.