Last updated May 14, 2013

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 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 51: In Section 1.3.6, the role of rows and columns should be reversed in “at least as many rows as columns”.
• Page 68: In the last display of Proposition 2.1.9, $2^{-m-1}$ should be $\frac{1}{100(m+1)^2}$.
• 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).
• p. 81: In Remark 2.2.2, “central limit” should be “central limit theorem”
• 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 183: In (2.127), the factor $1! \ldots (n-1)!$ is missing from the denominator.
• 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 $\sum_{j: |x_j| \leq \varepsilon^{100}} |x_j|^2 \geq \eps^{10}$ should be  $\sum_{j: |x_j| \leq \varepsilon} |x_j|^2 \geq \eta$, the final bound should be $\ll_{\eta,\delta} \varepsilon$ rather than $\ll \vaarepsilon$, 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 $n^{O_{\varepsilon,K}(1)}$ (two ocurrences), 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 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 Rex Cheung, Nick Cook, Jesus A Dominguez, Peter Forrester, Stephen Ge, Gautam Kamath, Fan Lau, Ilya Razenshteyn, Weiji Su and Ambuj Tewari for corrections.