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I’ve just finished writing the first draft of my second book coming out of the 2010 blog posts, namely “Topics in random matrix theory“, which was based primarily on my graduate course in the topic, though it also contains material from some additional posts related to random matrices on the blog.  It is available online here.  As usual, comments and corrections are welcome.  There is also a stub page for the book, which at present does not contain much more than the above link.

In this final set of lecture notes for this course, we leave the realm of self-adjoint matrix ensembles, such as Wigner random matrices, and consider instead the simplest examples of non-self-adjoint ensembles, namely the iid matrix ensembles. (I had also hoped to discuss recent progress in eigenvalue spacing distributions of Wigner matrices, but have run out of time. For readers interested in this topic, I can recommend the recent Bourbaki exposé of Alice Guionnet.)

The basic result in this area is

Theorem 1 (Circular law) Let ${M_n}$ be an ${n \times n}$ iid matrix, whose entries ${\xi_{ij}}$, ${1 \leq i,j \leq n}$ are iid with a fixed (complex) distribution ${\xi_{ij} \equiv \xi}$ of mean zero and variance one. Then the spectral measure ${\mu_{\frac{1}{\sqrt{n}}M_n}}$ converges both in probability and almost surely to the circular law ${\mu_{circ} := \frac{1}{\pi} 1_{|x|^2+|y|^2 \leq 1}\ dx dy}$, where ${x, y}$ are the real and imaginary coordinates of the complex plane.

This theorem has a long history; it is analogous to the semi-circular law, but the non-Hermitian nature of the matrices makes the spectrum so unstable that key techniques that are used in the semi-circular case, such as truncation and the moment method, no longer work; significant new ideas are required. In the case of random gaussian matrices, this result was established by Mehta (in the complex case) and by Edelman (in the real case), as was sketched out in Notes. In 1984, Girko laid out a general strategy for establishing the result for non-gaussian matrices, which formed the base of all future work on the subject; however, a key ingredient in the argument, namely a bound on the least singular value of shifts ${\frac{1}{\sqrt{n}} M_n - zI}$, was not fully justified at the time. A rigorous proof of the circular law was then established by Bai, assuming additional moment and boundedness conditions on the individual entries. These additional conditions were then slowly removed in a sequence of papers by Gotze-Tikhimirov, Girko, Pan-Zhou, and Tao-Vu, with the last moment condition being removed in a paper of myself, Van Vu, and Manjunath Krishnapur.

At present, the known methods used to establish the circular law for general ensembles rely very heavily on the joint independence of all the entries. It is a key challenge to see how to weaken this joint independence assumption.

Now we turn attention to another important spectral statistic, the least singular value ${\sigma_n(M)}$ of an ${n \times n}$ matrix ${M}$ (or, more generally, the least non-trivial singular value ${\sigma_p(M)}$ of a ${n \times p}$ matrix with ${p \leq n}$). This quantity controls the invertibility of ${M}$. Indeed, ${M}$ is invertible precisely when ${\sigma_n(M)}$ is non-zero, and the operator norm ${\|M^{-1}\|_{op}}$ of ${M^{-1}}$ is given by ${1/\sigma_n(M)}$. This quantity is also related to the condition number ${\sigma_1(M)/\sigma_n(M) = \|M\|_{op} \|M^{-1}\|_{op}}$ of ${M}$, which is of importance in numerical linear algebra. As we shall see in the next set of notes, the least singular value of ${M}$ (and more generally, of the shifts ${\frac{1}{\sqrt{n}} M - zI}$ for complex ${z}$) will be of importance in rigorously establishing the circular law for iid random matrices ${M}$, as it plays a key role in computing the Stieltjes transform ${\frac{1}{n} \hbox{tr} (\frac{1}{\sqrt{n}} M - zI)^{-1}}$ of such matrices, which as we have already seen is a powerful tool in understanding the spectra of random matrices.

The least singular value

$\displaystyle \sigma_n(M) = \inf_{\|x\|=1} \|Mx\|,$

which sits at the “hard edge” of the spectrum, bears a superficial similarity to the operator norm

$\displaystyle \|M\|_{op} = \sigma_1(M) = \sup_{\|x\|=1} \|Mx\|$

at the “soft edge” of the spectrum, that was discussed back in Notes 3, so one may at first think that the methods that were effective in controlling the latter, namely the epsilon-net argument and the moment method, would also work to control the former. The epsilon-net method does indeed have some effectiveness when dealing with rectangular matrices (in which the spectrum stays well away from zero), but the situation becomes more delicate for square matrices; it can control some “low entropy” portions of the infimum that arise from “structured” or “compressible” choices of ${x}$, but are not able to control the “generic” or “incompressible” choices of ${x}$, for which new arguments will be needed. As for the moment method, this can give the coarse order of magnitude (for instance, for rectangular matrices with ${p=yn}$ for ${0 < y < 1}$, it gives an upper bound of ${(1-\sqrt{y}+o(1))n}$ for the singular value with high probability, thanks to the Marchenko-Pastur law), but again this method begins to break down for square matrices, although one can make some partial headway by considering negative moments such as ${\hbox{tr} M^{-2}}$, though these are more difficult to compute than positive moments ${\hbox{tr} M^k}$.

So one needs to supplement these existing methods with additional tools. It turns out that the key issue is to understand the distance between one of the ${n}$ rows ${X_1,\ldots,X_n \in {\bf C}^n}$ of the matrix ${M}$, and the hyperplane spanned by the other ${n-1}$ rows. The reason for this is as follows. First suppose that ${\sigma_n(M)=0}$, so that ${M}$ is non-invertible, and there is a linear dependence between the rows ${X_1,\ldots,X_n}$. Thus, one of the ${X_i}$ will lie in the hyperplane spanned by the other rows, and so one of the distances mentioned above will vanish; in fact, one expects many of the ${n}$ distances to vanish. Conversely, whenever one of these distances vanishes, one has a linear dependence, and so ${\sigma_n(M)=0}$.

More generally, if the least singular value ${\sigma_n(M)}$ is small, one generically expects many of these ${n}$ distances to be small also, and conversely. Thus, control of the least singular value is morally equivalent to control of the distance between a row ${X_i}$ and the hyperplane spanned by the other rows. This latter quantity is basically the dot product of ${X_i}$ with a unit normal ${n_i}$ of this hyperplane.

When working with random matrices with jointly independent coefficients, we have the crucial property that the unit normal ${n_i}$ (which depends on all the rows other than ${X_i}$) is independent of ${X_i}$, so even after conditioning ${n_i}$ to be fixed, the entries of ${X_i}$ remain independent. As such, the dot product ${X_i \cdot n_i}$ is a familiar scalar random walk, and can be controlled by a number of tools, most notably Littlewood-Offord theorems and the Berry-Esséen central limit theorem. As it turns out, this type of control works well except in some rare cases in which the normal ${n_i}$ is “compressible” or otherwise highly structured; but epsilon-net arguments can be used to dispose of these cases. (This general strategy was first developed for the technically simpler singularity problem by Komlós, and then extended to the least singular value problem by Rudelson.)

These methods rely quite strongly on the joint independence on all the entries; it remains a challenge to extend them to more general settings. Even for Wigner matrices, the methods run into difficulty because of the non-independence of some of the entries (although it turns out one can understand the least singular value in such cases by rather different methods).

To simplify the exposition, we shall focus primarily on just one specific ensemble of random matrices, the Bernoulli ensemble ${M = (\xi_{ij})_{1 \leq i,j \leq n}}$ of random sign matrices, where ${\xi_{ij} = \pm 1}$ are independent Bernoulli signs. However, the results can extend to more general classes of random matrices, with the main requirement being that the coefficients are jointly independent.

Our study of random matrices, to date, has focused on somewhat general ensembles, such as iid random matrices or Wigner random matrices, in which the distribution of the individual entries of the matrices was essentially arbitrary (as long as certain moments, such as the mean and variance, were normalised). In these notes, we now focus on two much more special, and much more symmetric, ensembles:

• The Gaussian Unitary Ensemble (GUE), which is an ensemble of random ${n \times n}$ Hermitian matrices ${M_n}$ in which the upper-triangular entries are iid with distribution ${N(0,1)_{\bf C}}$, and the diagonal entries are iid with distribution ${N(0,1)_{\bf R}}$, and independent of the upper-triangular ones; and
• The Gaussian random matrix ensemble, which is an ensemble of random ${n \times n}$ (non-Hermitian) matrices ${M_n}$ whose entries are iid with distribution ${N(0,1)_{\bf C}}$.

The symmetric nature of these ensembles will allow us to compute the spectral distribution by exact algebraic means, revealing a surprising connection with orthogonal polynomials and with determinantal processes. This will, for instance, recover the semi-circular law for GUE, but will also reveal fine spacing information, such as the distribution of the gap between adjacent eigenvalues, which is largely out of reach of tools such as the Stieltjes transform method and the moment method (although the moment method, with some effort, is able to control the extreme edges of the spectrum).

Similarly, we will see for the first time the circular law for eigenvalues of non-Hermitian matrices.

There are a number of other highly symmetric ensembles which can also be treated by the same methods, most notably the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). However, for simplicity we shall focus just on the above two ensembles. For a systematic treatment of these ensembles, see the text by Deift.

In the foundations of modern probability, as laid out by Kolmogorov, the basic objects of study are constructed in the following order:

1. Firstly, one selects a sample space ${\Omega}$, whose elements ${\omega}$ represent all the possible states that one’s stochastic system could be in.
2. Then, one selects a ${\sigma}$-algebra ${{\mathcal B}}$ of events ${E}$ (modeled by subsets of ${\Omega}$), and assigns each of these events a probability ${{\bf P}(E) \in [0,1]}$ in a countably additive manner, so that the entire sample space has probability ${1}$.
3. Finally, one builds (commutative) algebras of random variables ${X}$ (such as complex-valued random variables, modeled by measurable functions from ${\Omega}$ to ${{\bf C}}$), and (assuming suitable integrability or moment conditions) one can assign expectations ${\mathop{\bf E} X}$ to each such random variable.

In measure theory, the underlying measure space ${\Omega}$ plays a prominent foundational role, with the measurable sets and measurable functions (the analogues of the events and the random variables) always being viewed as somehow being attached to that space. In probability theory, in contrast, it is the events and their probabilities that are viewed as being fundamental, with the sample space ${\Omega}$ being abstracted away as much as possible, and with the random variables and expectations being viewed as derived concepts. See Notes 0 for further discussion of this philosophy.

However, it is possible to take the abstraction process one step further, and view the algebra of random variables and their expectations as being the foundational concept, and ignoring both the presence of the original sample space, the algebra of events, or the probability measure.

There are two reasons for wanting to shed (or abstract away) these previously foundational structures. Firstly, it allows one to more easily take certain types of limits, such as the large ${n}$ limit ${n \rightarrow \infty}$ when considering ${n \times n}$ random matrices, because quantities built from the algebra of random variables and their expectations, such as the normalised moments of random matrices tend to be quite stable in the large ${n}$ limit (as we have seen in previous notes), even as the sample space and event space varies with ${n}$. (This theme of using abstraction to facilitate the taking of the large ${n}$ limit also shows up in the application of ergodic theory to combinatorics via the correspondence principle; see this previous blog post for further discussion.)

Secondly, this abstract formalism allows one to generalise the classical, commutative theory of probability to the more general theory of non-commutative probability theory, which does not have a classical underlying sample space or event space, but is instead built upon a (possibly) non-commutative algebra of random variables (or “observables”) and their expectations (or “traces”). This more general formalism not only encompasses classical probability, but also spectral theory (with matrices or operators taking the role of random variables, and the trace taking the role of expectation), random matrix theory (which can be viewed as a natural blend of classical probability and spectral theory), and quantum mechanics (with physical observables taking the role of random variables, and their expected value on a given quantum state being the expectation). It is also part of a more general “non-commutative way of thinking” (of which non-commutative geometry is the most prominent example), in which a space is understood primarily in terms of the ring or algebra of functions (or function-like objects, such as sections of bundles) placed on top of that space, and then the space itself is largely abstracted away in order to allow the algebraic structures to become less commutative. In short, the idea is to make algebra the foundation of the theory, as opposed to other possible choices of foundations such as sets, measures, categories, etc..

[Note that this foundational preference is to some extent a metamathematical one rather than a mathematical one; in many cases it is possible to rewrite the theory in a mathematically equivalent form so that some other mathematical structure becomes designated as the foundational one, much as probability theory can be equivalently formulated as the measure theory of probability measures. However, this does not negate the fact that a different choice of foundations can lead to a different way of thinking about the subject, and thus to ask a different set of questions and to discover a different set of proofs and solutions. Thus it is often of value to understand multiple foundational perspectives at once, to get a truly stereoscopic view of the subject.]

It turns out that non-commutative probability can be modeled using operator algebras such as ${C^*}$-algebras, von Neumann algebras, or algebras of bounded operators on a Hilbert space, with the latter being accomplished via the Gelfand-Naimark-Segal construction. We will discuss some of these models here, but just as probability theory seeks to abstract away its measure-theoretic models, the philosophy of non-commutative probability is also to downplay these operator algebraic models once some foundational issues are settled.

When one generalises the set of structures in one’s theory, for instance from the commutative setting to the non-commutative setting, the notion of what it means for a structure to be “universal”, “free”, or “independent” can change. The most familiar example of this comes from group theory. If one restricts attention to the category of abelian groups, then the “freest” object one can generate from two generators ${e,f}$ is the free abelian group of commutative words ${e^n f^m}$ with ${n,m \in {\bf Z}}$, which is isomorphic to the group ${{\bf Z}^2}$. If however one generalises to the non-commutative setting of arbitrary groups, then the “freest” object that can now be generated from two generators ${e,f}$ is the free group ${{\Bbb F}_2}$ of non-commutative words ${e^{n_1} f^{m_1} \ldots e^{n_k} f^{m_k}}$ with ${n_1,m_1,\ldots,n_k,m_k \in {\bf Z}}$, which is a significantly larger extension of the free abelian group ${{\bf Z}^2}$.

Similarly, when generalising classical probability theory to non-commutative probability theory, the notion of what it means for two or more random variables to be independent changes. In the classical (commutative) setting, two (bounded, real-valued) random variables ${X, Y}$ are independent if one has

$\displaystyle \mathop{\bf E} f(X) g(Y) = 0$

whenever ${f, g: {\bf R} \rightarrow {\bf R}}$ are well-behaved functions (such as polynomials) such that all of ${\mathop{\bf E} f(X)}$, ${\mathop{\bf E} g(Y)}$ vanishes. In the non-commutative setting, one can generalise the above definition to two commuting bounded self-adjoint variables; this concept is useful for instance in quantum probability, which is an abstraction of the theory of observables in quantum mechanics. But for two (bounded, self-adjoint) non-commutative random variables ${X, Y}$, the notion of classical independence no longer applies. As a substitute, one can instead consider the notion of being freely independent (or free for short), which means that

$\displaystyle \mathop{\bf E} f_1(X) g_1(Y) \ldots f_k(X) g_k(Y) = 0$

whenever ${f_1,g_1,\ldots,f_k,g_k: {\bf R} \rightarrow {\bf R}}$ are well-behaved functions such that all of ${\mathop{\bf E} f_1(X), \mathop{\bf E} g_1(Y), \ldots, \mathop{\bf E} f_k(X), \mathop{\bf E} g_k(Y)}$ vanish.

The concept of free independence was introduced by Voiculescu, and its study is now known as the subject of free probability. We will not attempt a systematic survey of this subject here; for this, we refer the reader to the surveys of Speicher and of Biane. Instead, we shall just discuss a small number of topics in this area to give the flavour of the subject only.

The significance of free probability to random matrix theory lies in the fundamental observation that random matrices which are independent in the classical sense, also tend to be independent in the free probability sense, in the large ${n}$ limit ${n \rightarrow \infty}$. (This is only possible because of the highly non-commutative nature of these matrices; as we shall see, it is not possible for non-trivial commuting independent random variables to be freely independent.) Because of this, many tedious computations in random matrix theory, particularly those of an algebraic or enumerative combinatorial nature, can be done more quickly and systematically by using the framework of free probability, which by design is optimised for algebraic tasks rather than analytical ones.

Much as free groups are in some sense “maximally non-commutative”, freely independent random variables are about as far from being commuting as possible. For instance, if ${X, Y}$ are freely independent and of expectation zero, then ${\mathop{\bf E} XYXY}$ vanishes, but ${\mathop{\bf E} XXYY}$ instead factors as ${(\mathop{\bf E} X^2) (\mathop{\bf E} Y^2)}$. As a consequence, the behaviour of freely independent random variables can be quite different from the behaviour of their classically independent commuting counterparts. Nevertheless there is a remarkably strong analogy between the two types of independence, in that results which are true in the classically independent case often have an interesting analogue in the freely independent setting. For instance, the central limit theorem (Notes 2) for averages of classically independent random variables, which roughly speaking asserts that such averages become gaussian in the large ${n}$ limit, has an analogue for averages of freely independent variables, the free central limit theorem, which roughly speaking asserts that such averages become semicircular in the large ${n}$ limit. One can then use this theorem to provide yet another proof of Wigner’s semicircle law (Notes 4).

Another important (and closely related) analogy is that while the distribution of sums of independent commutative random variables can be quickly computed via the characteristic function (i.e. the Fourier transform of the distribution), the distribution of sums of freely independent non-commutative random variables can be quickly computed using the Stieltjes transform instead (or with closely related objects, such as the ${R}$-transform of Voiculescu). This is strongly reminiscent of the appearance of the Stieltjes transform in random matrix theory, and indeed we will see many parallels between the use of the Stieltjes transform here and in Notes 4.

As mentioned earlier, free probability is an excellent tool for computing various expressions of interest in random matrix theory, such as asymptotic values of normalised moments in the large ${n}$ limit ${n \rightarrow \infty}$. Nevertheless, as it only covers the asymptotic regime in which ${n}$ is sent to infinity while holding all other parameters fixed, there are some aspects of random matrix theory to which the tools of free probability are not sufficient by themselves to resolve (although it can be possible to combine free probability theory with other tools to then answer these questions). For instance, questions regarding the rate of convergence of normalised moments as ${n \rightarrow \infty}$ are not directly answered by free probability, though if free probability is combined with tools such as concentration of measure (Notes 1) then such rate information can often be recovered. For similar reasons, free probability lets one understand the behaviour of ${k^{th}}$ moments as ${n \rightarrow \infty}$ for fixed ${k}$, but has more difficulty dealing with the situation in which ${k}$ is allowed to grow slowly in ${n}$ (e.g. ${k = O(\log n)}$). Because of this, free probability methods are effective at controlling the bulk of the spectrum of a random matrix, but have more difficulty with the edges of that spectrum (as well as with related concepts such as the operator norm, Notes 3) as well as with fine-scale structure of the spectrum. Finally, free probability methods are most effective when dealing with matrices that are Hermitian with bounded operator norm, largely because the spectral theory of bounded self-adjoint operators in the infinite-dimensional setting of the large ${n}$ limit is non-pathological. (This is ultimately due to the stable nature of eigenvalues in the self-adjoint setting; see this previous blog post for discussion.) For non-self-adjoint operators, free probability needs to be augmented with additional tools, most notably by bounds on least singular values, in order to recover the required stability for the various spectral data of random matrices to behave continuously with respect to the large ${n}$ limit. We will discuss this latter point in a later set of notes.

We can now turn attention to one of the centerpiece universality results in random matrix theory, namely the Wigner semi-circle law for Wigner matrices. Recall from previous notes that a Wigner Hermitian matrix ensemble is a random matrix ensemble ${M_n = (\xi_{ij})_{1 \leq i,j \leq n}}$ of Hermitian matrices (thus ${\xi_{ij} = \overline{\xi_{ji}}}$; this includes real symmetric matrices as an important special case), in which the upper-triangular entries ${\xi_{ij}}$, ${i>j}$ are iid complex random variables with mean zero and unit variance, and the diagonal entries ${\xi_{ii}}$ are iid real variables, independent of the upper-triangular entries, with bounded mean and variance. Particular special cases of interest include the Gaussian Orthogonal Ensemble (GOE), the symmetric random sign matrices (aka symmetric Bernoulli ensemble), and the Gaussian Unitary Ensemble (GUE).

In previous notes we saw that the operator norm of ${M_n}$ was typically of size ${O(\sqrt{n})}$, so it is natural to work with the normalised matrix ${\frac{1}{\sqrt{n}} M_n}$. Accordingly, given any ${n \times n}$ Hermitian matrix ${M_n}$, we can form the (normalised) empirical spectral distribution (or ESD for short)

$\displaystyle \mu_{\frac{1}{\sqrt{n}} M_n} := \frac{1}{n} \sum_{j=1}^n \delta_{\lambda_j(M_n) / \sqrt{n}},$

of ${M_n}$, where ${\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)}$ are the (necessarily real) eigenvalues of ${M_n}$, counting multiplicity. The ESD is a probability measure, which can be viewed as a distribution of the normalised eigenvalues of ${M_n}$.

When ${M_n}$ is a random matrix ensemble, then the ESD ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ is now a random measure – i.e. a random variable taking values in the space ${\hbox{Pr}({\mathbb R})}$ of probability measures on the real line. (Thus, the distribution of ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ is a probability measure on probability measures!)

Now we consider the behaviour of the ESD of a sequence of Hermitian matrix ensembles ${M_n}$ as ${n \rightarrow \infty}$. Recall from Notes 0 that for any sequence of random variables in a ${\sigma}$-compact metrisable space, one can define notions of convergence in probability and convergence almost surely. Specialising these definitions to the case of random probability measures on ${{\mathbb R}}$, and to deterministic limits, we see that a sequence of random ESDs ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ converge in probability (resp. converge almost surely) to a deterministic limit ${\mu \in \hbox{Pr}({\mathbb R})}$ (which, confusingly enough, is a deterministic probability measure!) if, for every test function ${\varphi \in C_c({\mathbb R})}$, the quantities ${\int_{\mathbb R} \varphi\ d\mu_{\frac{1}{\sqrt{n}} M_n}}$ converge in probability (resp. converge almost surely) to ${\int_{\mathbb R} \varphi\ d\mu}$.

Remark 1 As usual, convergence almost surely implies convergence in probability, but not vice versa. In the special case of random probability measures, there is an even weaker notion of convergence, namely convergence in expectation, defined as follows. Given a random ESD ${\mu_{\frac{1}{\sqrt{n}} M_n}}$, one can form its expectation ${{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n} \in \hbox{Pr}({\mathbb R})}$, defined via duality (the Riesz representation theorem) as

$\displaystyle \int_{\mathbb R} \varphi\ d{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n} := {\bf E} \int_{\mathbb R} \varphi\ d \mu_{\frac{1}{\sqrt{n}} M_n};$

this probability measure can be viewed as the law of a random eigenvalue ${\frac{1}{\sqrt{n}}\lambda_i(M_n)}$ drawn from a random matrix ${M_n}$ from the ensemble. We then say that the ESDs converge in expectation to a limit ${\mu \in \hbox{Pr}({\mathbb R})}$ if ${{\bf E} \mu_{\frac{1}{\sqrt{n}} M_n}}$ converges the vague topology to ${\mu}$, thus

$\displaystyle {\bf E} \int_{\mathbb R} \varphi\ d \mu_{\frac{1}{\sqrt{n}} M_n} \rightarrow \int_{\mathbb R} \varphi\ d\mu$

for all ${\phi \in C_c({\mathbb R})}$.

In general, these notions of convergence are distinct from each other; but in practice, one often finds in random matrix theory that these notions are effectively equivalent to each other, thanks to the concentration of measure phenomenon.

Exercise 1 Let ${M_n}$ be a sequence of ${n \times n}$ Hermitian matrix ensembles, and let ${\mu}$ be a continuous probability measure on ${{\mathbb R}}$.

• Show that ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ converges almost surely to ${\mu}$ if and only if ${\mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)}$ converges almost surely to ${\mu(-\infty,\lambda)}$ for all ${\lambda \in {\mathbb R}}$.
• Show that ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ converges in probability to ${\mu}$ if and only if ${\mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)}$ converges in probability to ${\mu(-\infty,\lambda)}$ for all ${\lambda \in {\mathbb R}}$.
• Show that ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ converges in expectation to ${\mu}$ if and only if ${\mathop{\mathbb E} \mu_{\frac{1}{\sqrt{n}}}(-\infty,\lambda)}$ converges to ${\mu(-\infty,\lambda)}$ for all ${\lambda \in {\mathbb R}}$.

We can now state the Wigner semi-circular law.

Theorem 1 (Semicircular law) Let ${M_n}$ be the top left ${n \times n}$ minors of an infinite Wigner matrix ${(\xi_{ij})_{i,j \geq 1}}$. Then the ESDs ${\mu_{\frac{1}{\sqrt{n}} M_n}}$ converge almost surely (and hence also in probability and in expectation) to the Wigner semi-circular distribution

$\displaystyle \mu_{sc} := \frac{1}{2\pi} (4-|x|^2)^{1/2}_+\ dx. \ \ \ \ \ (1)$

A numerical example of this theorem in action can be seen at the MathWorld entry for this law.

The semi-circular law nicely complements the upper Bai-Yin theorem from Notes 3, which asserts that (in the case when the entries have finite fourth moment, at least), the matrices ${\frac{1}{\sqrt{n}} M_n}$ almost surely has operator norm at most ${2+o(1)}$. Note that the operator norm is the same thing as the largest magnitude of the eigenvalues. Because the semi-circular distribution (1) is supported on the interval ${[-2,2]}$ with positive density on the interior of this interval, Theorem 1 easily supplies the lower Bai-Yin theorem, that the operator norm of ${\frac{1}{\sqrt{n}} M_n}$ is almost surely at least ${2-o(1)}$, and thus (in the finite fourth moment case) the norm is in fact equal to ${2+o(1)}$. Indeed, we have just shown that the circular law provides an alternate proof of the lower Bai-Yin bound (Proposition 11 of Notes 3).

As will hopefully become clearer in the next set of notes, the semi-circular law is the noncommutative (or free probability) analogue of the central limit theorem, with the semi-circular distribution (1) taking on the role of the normal distribution. Of course, there is a striking difference between the two distributions, in that the former is compactly supported while the latter is merely subgaussian. One reason for this is that the concentration of measure phenomenon is more powerful in the case of ESDs of Wigner matrices than it is for averages of iid variables; compare the concentration of measure results in Notes 3 with those in Notes 1.

There are several ways to prove (or at least to heuristically justify) the circular law. In this set of notes we shall focus on the two most popular methods, the moment method and the Stieltjes transform method, together with a third (heuristic) method based on Dyson Brownian motion (Notes 3b). In the next set of notes we shall also study the free probability method, and in the set of notes after that we use the determinantal processes method (although this method is initially only restricted to highly symmetric ensembles, such as GUE).

One theme in this course will be the central nature played by the gaussian random variables ${X \equiv N(\mu,\sigma^2)}$. Gaussians have an incredibly rich algebraic structure, and many results about general random variables can be established by first using this structure to verify the result for gaussians, and then using universality techniques (such as the Lindeberg exchange strategy) to extend the results to more general variables.

One way to exploit this algebraic structure is to continuously deform the variance ${t := \sigma^2}$ from an initial variance of zero (so that the random variable is deterministic) to some final level ${T}$. We would like to use this to give a continuous family ${t \mapsto X_t}$ of random variables ${X_t \equiv N(\mu, t)}$ as ${t}$ (viewed as a “time” parameter) runs from ${0}$ to ${T}$.

At present, we have not completely specified what ${X_t}$ should be, because we have only described the individual distribution ${X_t \equiv N(\mu,t)}$ of each ${X_t}$, and not the joint distribution. However, there is a very natural way to specify a joint distribution of this type, known as Brownian motion. In these notes we lay the necessary probability theory foundations to set up this motion, and indicate its connection with the heat equation, the central limit theorem, and the Ornstein-Uhlenbeck process. This is the beginning of stochastic calculus, which we will not develop fully here.

We will begin with one-dimensional Brownian motion, but it is a simple matter to extend the process to higher dimensions. In particular, we can define Brownian motion on vector spaces of matrices, such as the space of ${n \times n}$ Hermitian matrices. This process is equivariant with respect to conjugation by unitary matrices, and so we can quotient out by this conjugation and obtain a new process on the quotient space, or in other words on the spectrum of ${n \times n}$ Hermitian matrices. This process is called Dyson Brownian motion, and turns out to have a simple description in terms of ordinary Brownian motion; it will play a key role in several of the subsequent notes in this course.

Let ${A}$ be a Hermitian ${n \times n}$ matrix. By the spectral theorem for Hermitian matrices (which, for sake of completeness, we prove below), one can diagonalise ${A}$ using a sequence

$\displaystyle \lambda_1(A) \geq \ldots \geq \lambda_n(A)$

of ${n}$ real eigenvalues, together with an orthonormal basis of eigenvectors ${u_1(A),\ldots,u_n(A) \in {\mathbb C}^n}$. (The eigenvalues are uniquely determined by ${A}$, but the eigenvectors have a little ambiguity to them, particularly if there are repeated eigenvalues; for instance, one could multiply each eigenvector by a complex phase ${e^{i\theta}}$. In these notes we are arranging eigenvalues in descending order; of course, one can also arrange eigenvalues in increasing order, which causes some slight notational changes in the results below.) The set ${\{\lambda_1(A),\ldots,\lambda_n(A)\}}$ is known as the spectrum of ${A}$.

A basic question in linear algebra asks the extent to which the eigenvalues ${\lambda_1(A),\ldots,\lambda_n(A)}$ and ${\lambda_1(B),\ldots,\lambda_n(B)}$ of two Hermitian matrices ${A, B}$ constrains the eigenvalues ${\lambda_1(A+B),\ldots,\lambda_n(A+B)}$ of the sum. For instance, the linearity of trace

$\displaystyle \hbox{tr}(A+B) = \hbox{tr}(A)+\hbox{tr}(B),$

when expressed in terms of eigenvalues, gives the trace constraint

$\displaystyle \lambda_1(A+B)+\ldots+\lambda_n(A+B) = \lambda_1(A)+\ldots+\lambda_n(A) \ \ \ \ \ (1)$

$\displaystyle +\lambda_1(B)+\ldots+\lambda_n(B);$

the identity

$\displaystyle \lambda_1(A) = \sup_{|v|=1} v^* Av \ \ \ \ \ (2)$

(together with the counterparts for ${B}$ and ${A+B}$) gives the inequality

$\displaystyle \lambda_1(A+B) \leq \lambda_1(A) + \lambda_1(B); \ \ \ \ \ (3)$

and so forth.

The complete answer to this problem is a fascinating one, requiring a strangely recursive description (once known as Horn’s conjecture, which is now solved), and connected to a large number of other fields of mathematics, such as geometric invariant theory, intersection theory, and the combinatorics of a certain gadget known as a “honeycomb”. See for instance my survey with Allen Knutson on this topic some years ago.

In typical applications to random matrices, one of the matrices (say, ${B}$) is “small” in some sense, so that ${A+B}$ is a perturbation of ${A}$. In this case, one does not need the full strength of the above theory, and instead rely on a simple aspect of it pointed out by Helmke and Rosenthal and by Totaro, which generates several of the eigenvalue inequalities relating ${A}$, ${B}$, and ${C}$, of which (1) and (3) are examples. (Actually, this method eventually generates all of the eigenvalue inequalities, but this is a non-trivial fact to prove.) These eigenvalue inequalities can mostly be deduced from a number of minimax characterisations of eigenvalues (of which (2) is a typical example), together with some basic facts about intersections of subspaces. Examples include the Weyl inequalities

$\displaystyle \lambda_{i+j-1}(A+B) \leq \lambda_i(A) + \lambda_j(B), \ \ \ \ \ (4)$

valid whenever ${i,j \geq 1}$ and ${i+j-1 \leq n}$, and the Ky Fan inequality

$\displaystyle \lambda_1(A+B)+\ldots+\lambda_k(A+B) \leq$

$\displaystyle \lambda_1(A)+\ldots+\lambda_k(A) + \lambda_1(B)+\ldots+\lambda_k(B). \ \ \ \ \ (5)$

One consequence of these inequalities is that the spectrum of a Hermitian matrix is stable with respect to small perturbations.

We will also establish some closely related inequalities concerning the relationships between the eigenvalues of a matrix, and the eigenvalues of its minors.

Many of the inequalities here have analogues for the singular values of non-Hermitian matrices (which is consistent with the discussion near Exercise 16 of Notes 3). However, the situation is markedly different when dealing with eigenvalues of non-Hermitian matrices; here, the spectrum can be far more unstable, if pseudospectrum is present. Because of this, the theory of the eigenvalues of a random non-Hermitian matrix requires an additional ingredient, namely upper bounds on the prevalence of pseudospectrum, which after recentering the matrix is basically equivalent to establishing lower bounds on least singular values. We will discuss this point in more detail in later notes.

We will work primarily here with Hermitian matrices, which can be viewed as self-adjoint transformations on complex vector spaces such as ${{\mathbb C}^n}$. One can of course specialise the discussion to real symmetric matrices, in which case one can restrict these complex vector spaces to their real counterparts ${{\mathbb R}^n}$. The specialisation of the complex theory below to the real case is straightforward and is left to the interested reader.

Now that we have developed the basic probabilistic tools that we will need, we now turn to the main subject of this course, namely the study of random matrices. There are many random matrix models (aka matrix ensembles) of interest – far too many to all be discussed in a single course. We will thus focus on just a few simple models. First of all, we shall restrict attention to square matrices ${M = (\xi_{ij})_{1 \leq i,j \leq n}}$, where ${n}$ is a (large) integer and the ${\xi_{ij}}$ are real or complex random variables. (One can certainly study rectangular matrices as well, but for simplicity we will only look at the square case.) Then, we shall restrict to three main models:

• Iid matrix ensembles, in which the coefficients ${\xi_{ij}}$ are iid random variables with a single distribution ${\xi_{ij} \equiv \xi}$. We will often normalise ${\xi}$ to have mean zero and unit variance. Examples of iid models include the Bernouli ensemble (aka random sign matrices) in which the ${\xi_{ij}}$ are signed Bernoulli variables, the real gaussian matrix ensemble in which ${\xi_{ij} \equiv N(0,1)_{\bf R}}$, and the complex gaussian matrix ensemble in which ${\xi_{ij} \equiv N(0,1)_{\bf C}}$.
• Symmetric Wigner matrix ensembles, in which the upper triangular coefficients ${\xi_{ij}}$, ${j \geq i}$ are jointly independent and real, but the lower triangular coefficients ${\xi_{ij}}$, ${j are constrained to equal their transposes: ${\xi_{ij}=\xi_{ji}}$. Thus ${M}$ by construction is always a real symmetric matrix. Typically, the strictly upper triangular coefficients will be iid, as will the diagonal coefficients, but the two classes of coefficients may have a different distribution. One example here is the symmetric Bernoulli ensemble, in which both the strictly upper triangular and the diagonal entries are signed Bernoulli variables; another important example is the Gaussian Orthogonal Ensemble (GOE), in which the upper triangular entries have distribution ${N(0,1)_{\bf R}}$ and the diagonal entries have distribution ${N(0,2)_{\bf R}}$. (We will explain the reason for this discrepancy later.)
• Hermitian Wigner matrix ensembles, in which the upper triangular coefficients are jointly independent, with the diagonal entries being real and the strictly upper triangular entries complex, and the lower triangular coefficients ${\xi_{ij}}$, ${j are constrained to equal their adjoints: ${\xi_{ij} = \overline{\xi_{ji}}}$. Thus ${M}$ by construction is always a Hermitian matrix. This class of ensembles contains the symmetric Wigner ensembles as a subclass. Another very important example is the Gaussian Unitary Ensemble (GUE), in which all off-diagional entries have distribution ${N(0,1)_{\bf C}}$, but the diagonal entries have distribution ${N(0,1)_{\bf R}}$.

Given a matrix ensemble ${M}$, there are many statistics of ${M}$ that one may wish to consider, e.g. the eigenvalues or singular values of ${M}$, the trace and determinant, etc. In these notes we will focus on a basic statistic, namely the operator norm

$\displaystyle \| M \|_{op} := \sup_{x \in {\bf C}^n: |x|=1} |Mx| \ \ \ \ \ (1)$

of the matrix ${M}$. This is an interesting quantity in its own right, but also serves as a basic upper bound on many other quantities. (For instance, ${\|M\|_{op}}$ is also the largest singular value ${\sigma_1(M)}$ of ${M}$ and thus dominates the other singular values; similarly, all eigenvalues ${\lambda_i(M)}$ of ${M}$ clearly have magnitude at most ${\|M\|_{op}}$.) Because of this, it is particularly important to get good upper tail bounds

$\displaystyle {\bf P}( \|M\|_{op} \geq \lambda ) \leq \ldots$

on this quantity, for various thresholds ${\lambda}$. (Lower tail bounds are also of interest, of course; for instance, they give us confidence that the upper tail bounds are sharp.) Also, as we shall see, the problem of upper bounding ${\|M\|_{op}}$ can be viewed as a non-commutative analogue of upper bounding the quantity ${|S_n|}$ studied in Notes 1. (The analogue of the central limit theorem in Notes 2 is the Wigner semi-circular law, which will be studied in the next set of notes.)

An ${n \times n}$ matrix consisting entirely of ${1}$s has an operator norm of exactly ${n}$, as can for instance be seen from the Cauchy-Schwarz inequality. More generally, any matrix whose entries are all uniformly ${O(1)}$ will have an operator norm of ${O(n)}$ (which can again be seen from Cauchy-Schwarz, or alternatively from Schur’s test, or from a computation of the Frobenius norm). However, this argument does not take advantage of possible cancellations in ${M}$. Indeed, from analogy with concentration of measure, when the entries of the matrix ${M}$ are independent, bounded and have mean zero, we expect the operator norm to be of size ${O(\sqrt{n})}$ rather than ${O(n)}$. We shall see shortly that this intuition is indeed correct. (One can see, though, that the mean zero hypothesis is important; from the triangle inequality we see that if we add the all-ones matrix (for instance) to a random matrix with mean zero, to obtain a random matrix whose coefficients all have mean ${1}$, then at least one of the two random matrices necessarily has operator norm at least ${n/2}$.)

As mentioned before, there is an analogy here with the concentration of measure phenomenon, and many of the tools used in the latter (e.g. the moment method) will also appear here. (Indeed, we will be able to use some of the concentration inequalities from Notes 1 directly to help control ${\|M\|_{op}}$ and related quantities.) Similarly, just as many of the tools from concentration of measure could be adapted to help prove the central limit theorem, several the tools seen here will be of use in deriving the semi-circular law.

The most advanced knowledge we have on the operator norm is given by the Tracy-Widom law, which not only tells us where the operator norm is concentrated in (it turns out, for instance, that for a Wigner matrix (with some additional technical assumptions), it is concentrated in the range ${[2\sqrt{n} - O(n^{-1/6}), 2\sqrt{n} + O(n^{-1/6})]}$), but what its distribution in that range is. While the methods in this set of notes can eventually be pushed to establish this result, this is far from trivial, and will only be briefly discussed here. (We may return to the Tracy-Widom law later in this course, though.)

Consider the sum ${S_n := X_1+\ldots+X_n}$ of iid real random variables ${X_1,\ldots,X_n \equiv X}$ of finite mean ${\mu}$ and variance ${\sigma^2}$ for some ${\sigma > 0}$. Then the sum ${S_n}$ has mean ${n\mu}$ and variance ${n\sigma^2}$, and so (by Chebyshev’s inequality) we expect ${S_n}$ to usually have size ${n\mu + O(\sqrt{n} \sigma)}$. To put it another way, if we consider the normalised sum

$\displaystyle Z_n := \frac{S_n - n \mu}{\sqrt{n} \sigma} \ \ \ \ \ (1)$

then ${Z_n}$ has been normalised to have mean zero and variance ${1}$, and is thus usually of size ${O(1)}$.

In the previous set of notes, we were able to establish various tail bounds on ${Z_n}$. For instance, from Chebyshev’s inequality one has

$\displaystyle {\bf P}(|Z_n| > \lambda) \leq \lambda^{-2}, \ \ \ \ \ (2)$

and if the original distribution ${X}$ was bounded or subgaussian, we had the much stronger Chernoff bound

$\displaystyle {\bf P}(|Z_n| > \lambda) \leq C \exp( - c \lambda^2 ) \ \ \ \ \ (3)$

for some absolute constants ${C, c > 0}$; in other words, the ${Z_n}$ are uniformly subgaussian.

Now we look at the distribution of ${Z_n}$. The fundamental central limit theorem tells us the asymptotic behaviour of this distribution:

Theorem 1 (Central limit theorem) Let ${X_1,\ldots,X_n \equiv X}$ be iid real random variables of finite mean ${\mu}$ and variance ${\sigma^2}$ for some ${\sigma > 0}$, and let ${Z_n}$ be the normalised sum (1). Then as ${n \rightarrow \infty}$, ${Z_n}$ converges in distribution to the standard normal distribution ${N(0,1)_{\bf R}}$.

Exercise 1 Show that ${Z_n}$ does not converge in probability or in the almost sure sense (in the latter case, we think of ${X_1,X_2,\ldots}$ as an infinite sequence of iid random variables). (Hint: the intuition here is that for two very different values ${n_1 \ll n_2}$ of ${n}$, the quantities ${Z_{n_1}}$ and ${Z_{n_2}}$ are almost independent of each other, since the bulk of the sum ${S_{n_2}}$ is determined by those ${X_n}$ with ${n > n_1}$. Now make this intuition precise.)

Exercise 2 Use Stirling’s formula from Notes 0a to verify the central limit theorem in the case when ${X}$ is a Bernoulli distribution, taking the values ${0}$ and ${1}$ only. (This is a variant of Exercise 2 from those notes, or Exercise 2 from Notes 1. It is easy to see that once one does this, one can rescale and handle any other two-valued distribution also.)

Exercise 3 Use Exercise 9 from Notes 1 to verify the central limit theorem in the case when ${X}$ is gaussian.

Note we are only discussing the case of real iid random variables. The case of complex random variables (or more generally, vector-valued random variables) is a little bit more complicated, and will be discussed later in this post.

The central limit theorem (and its variants, which we discuss below) are extremely useful tools in random matrix theory, in particular through the control they give on random walks (which arise naturally from linear functionals of random matrices). But the central limit theorem can also be viewed as a “commutative” analogue of various spectral results in random matrix theory (in particular, we shall see in later lectures that the Wigner semicircle law can be viewed in some sense as a “noncommutative” or “free” version of the central limit theorem). Because of this, the techniques used to prove the central limit theorem can often be adapted to be useful in random matrix theory. Because of this, we shall use these notes to dwell on several different proofs of the central limit theorem, as this provides a convenient way to showcase some of the basic methods that we will encounter again (in a more sophisticated form) when dealing with random matrices.