Last updated October 27, 2014

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 $X_N$.
Errata:
• Page 16: In Example 1.1.19, the Poisson distribution should be classified as having sub-exponential tail rather than being sub-Gaussian.
• 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 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 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 (vi), (vii), 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).
• Page 68: In the last display of Proposition 2.1.9, $2^{-m-1}$ should be $\frac{1}{100(m+1)^2}$.
• 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 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).
• Page 81: In Remark 2.2.2, “central limit” should be “central limit theorem”
• 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 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 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.
• 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.
• Page 183: In (2.127), the factor $1! \ldots (n-1)!$ is missing from the denominator.
• 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 192: In Footnote 52 to Section 2.6.3, the exponent $2$ should be $1/2$ instead.
• Page 203: In Exercise 2.6.6, a factor of $n^{-1/2}$ is missing in the $O()$ error term.  Earlier in the eigenfunction equation for $\tilde \phi_m$, $L_{1/\sqrt{n}}$ should be $L_{1/n}$.
• 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 $j$.
• 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 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 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 $(n-1)!$.
Thanks to Sviatoslav Archava, Rex Cheung, Nick Cook, Jesus A Dominguez, Peter Forrester, Stephen Ge, Gautam Kamath, Fan Lau, Travis Martin, Doron Puder, Ilya Razenshteyn, Frieder Simon, Weiji Su, Ambuj Tewari, Dominik Tomecki, and Felix V. for corrections.