Note to self: Upper Half-Plane Fixed Point Theorems

Table of Contents

• Table of Contents
• Motivation
• Carathéodory–Riffen–Finsler Pseudometric
  • Definition
  • Schwarz–Pick Lemma for the CRF Pseudometric
• The Poincaré Metric
  • On the Unit Disk
  • On the Upper Half-Plane
• Fixed-Point Theorems
  • Denjoy–Wolff Theorem
• Complex Implicit Function Theorem
• Schwarz Reflection Principle
• References

Motivation

If you’ve spent any time with random matrix theory like I have recently you will almost certainly run into something called a deterministic equivalent. These often show up as functions mapping the upper half of the complex plane \(\mathbb{H} = \{ z \in \mathbb{C} : \mathrm{Im}(z) > 0 \}\) into itself that (uniquely) solve some fixed-point equation.

To give a concrete example, suppose we take a random matrix of the form \(X = C^{1/2} Z\), where:

• \(C \in \mathbb{R}^{p \times p}\) is a deterministic positive semidefinite matrix with bounded operator norm,
• \(Z \in \mathbb{R}^{p \times n}\) has independent, zero-mean, unit-variance entries, and satisfies some mild moment conditions (typically something like bounded fourth moment or sub-Gaussian tails).

We can define the Stieltje transform of the empirical spectral distribution of the eigenvalues of \(\frac{1}{n} XX^\top\) as

where \(z \in \mathbb{H}\) is a spectral parameter, and \(\lambda_i\) are the eigenvalues of \(\frac{1}{n}XX^\top\).

It is a well known results that in the so-called thermodynamic limit (when both \(p, n \to \infty\) and \(p/n \to c\) for some constant \(c > 0\)) that \(s_n(z)\) is asymptotically well approximated by a deterministic function \(s(z)\) which takes the form (see Silverstein and Bai¹ for a classical reference)

where \(m(z):\mathbb{H}\mapsto \mathbb{H}\) uniquely solves the fixed-point equation

This deterministic function \(m(z)\) is a key object since it lets us describe the limiting behavior of the spectrum and other spectral quantities of interest. However, \(m(z)\) generally cannot be written in a closed-form.

Nonetheless, we can still say some things about it. For example, we can show that it is analytic on \(\mathbb{H}\), and we can approximate it numerically by iterating a fixed-point map. These properties (and more) follow from classical results in complex analysis but adapted from the unit disk to the upper half-plane. In this post I just want to highlight a few of these results, and how they apply in this setting.

Carathéodory–Riffen–Finsler Pseudometric

Before we state the Schwarz–Pick lemma for the Poincaré metric, we’ll need a couple of definitions. I will present a slightly more general metric called the Carathéodory–Riffen–Finsler (CRF) pseudometric. We refer to the original papers by Harris ^{2 3} for a more detailed discussion of this metric and its properties. What follows is adapted from my master’s thesis.

Definition

Let \(D\) be a domain in a normed linear space over the complex numbers \(X\). We define the infinitesimal Carathéodory–Riffen–Finsler (CRF) pseudometric as

where \(\mathbb{D}\) denotes the open complex unit disk of radius 1 and \(\mathrm{Hol}(D, \mathbb{D})\) is the set of holomorphic functions from \(D\) to \(\mathbb{D}\). The pseudometric \(\alpha\) is an infinitesimal Finsler pseudometric on \(D\), meaning that it is non-negative, lower semicontinuous, locally bounded, and satisfies

for all \((x, v) \in D \times X\) and \(t \in \mathbb{R}\).

Let \(\Gamma\) be the set of all curves in \(D\) with piecewise continuous derivatives (called admissible curves), and define

The pseudometric \(\alpha\) is a seminorm at each point in \(D\), and \(\mathcal{L}(\gamma)\) is interpreted as the length of the curve \(\gamma\) with respect to \(\alpha\). Then, the CRF pseudometric \(\rho\) on \(D\) is then defined by

As the name suggests, \(\rho\) is generally a pseudometric. For instance, if we let \(X=D=\mathbb{C}\), then any holomorphic function \(f : \mathbb{C} \to \mathbb{D}\) is a constant function by Liouville’s theorem. This means that the CRF pseudometric is identically zero on the whole complex plane. On the other hand, a sufficient condition for the CRF pseudometric to be a metric is that the domain \(D\) is bounded, in which case it is possible to show that

where \(d\) is the diameter of \(D\). By extension, if there exists a biholomorphic map \(f : D_1 \to D_2\) between two a domain \(D_1\) and a bounded domain \(D_2\), then the CRF pseudometric on \(D_1\) is also a metric by biholomorphic invariance (which we discuss below). This will ensure that the CRF pseudometric is a metric in the cases of the unit disk and the upper half-plane for instance.

Schwarz–Pick Lemma for the CRF Pseudometric

We can state a version of the Schwarz–Pick lemma that holds for the CRF pseudometric.

Proposition 1 tells us that the CRF pseudometric is non-expansive for holomorphic maps. In particular, if we apply it to a biholomorphic map \(f : D_1 \to D_2\), we get

which means the inequalities must be equalities and

for all \(x, y \in D_1\). In other words, the CRF pseudometric is biholomorphically invariant ⁴.

There is also an infinitesimal version of the Schwarz–Pick lemma, which is just the derivative form of Proposition 1. We state it here for completeness, but before we do so let us first highlight the relationship between the infinitesimal CRF pseudometric and the CRF pseudometric.

Fix a point \(x \in D\) and consider a the smooth curve \(\gamma(t) = x+tv\) with tangent vector \(v\) at \(t=0\). We suppose that \(t\) is small enough so that \(\gamma(s) \in D\) for all \(s \in [0, t]\). By definition of the CRF pseudometric,

Taking the limit as \(t \to 0^{+}\),

This gives us one direction of the inequality. For the other direction, let \(\epsilon > 0\) and choose \(f_0\in \mathrm{Hol}(D, \mathbb{D})\) such that

Choose \(\delta > 0\) small enough sucht that \(\|f_0(x+tv) - f_0(x) - tDf_0(x)v\| < \epsilon t\) whenever \(t\|v\| = \|x+tv - x\| < \delta\). Then,

By the integrated form of the Schwarz–Pick lemma (Proposition 1),

where we have used the explicit form of the Poincaré metric on the unit disk (see below). Dividing both sides by \(t\) and taking the limit as \(t \to 0^{+}\) gives us

Since \(\epsilon\) is arbitrary, we conclude that the integrated form of the CRF pseudometric is related to the infinitesimal CRF pseudometric through the Schwarz–Pick lemma as

We can now state the infinitesimal version of the Schwarz–Pick lemma.

The Poincaré Metric

We can now specialize the CRF pseudometric to the unit disk and the upper half-plane.

On the Unit Disk

As hinted above, the CRF pseudometric is a generalization of the Poincaré metric on the open unit disk \(\mathbb{D}\), which plays a central role in complex analysis. If we denote by \(\rho_{\mathbb{D}}\) the CRF pseudometric on the open unit disk, then we have the explicit formula

The Schwarz–Pick lemma for the CRF pseudometric (Proposition 1) and the infinitesimal version (Proposition 2) can be specialized to the Poincaré metric. In this case, let \(f : \mathbb{D} \to \mathbb{D}\) be a holomorphic function. Then for all \(z_1, z_2 \in \mathbb{D}\),

In particular, since \(x\mapsto \tanh(x)\) is an increasing function, we can also write this equivalently as

For \(z_1=z\) and \(z_2=z+h\), the above implies

Taking the limit as \(h \to 0\) using the fact that \(f\) is holomorphic and rearranging,

for all \(z \in \mathbb{D}\). This is the differential form of the Schwarz–Pick lemma for the Poincaré metric.

In the case of the Poincaré metric on the unit disk, the Schwarz–Pick lemma tells us more than just non-expansiveness. It also comes with a rigidity statement: if equality holds in either the distance form or the differential form of the inequality, then the function must be a Möbius automorphism of the disk. That is, if

for some distinct \(z_1, z_2 \in \mathbb{D}\), or if

for some \(z \in \mathbb{D}\), then \(f\) must be of the form

Let us collect these results in a single proposition. We refer to standard references for details of the proof ⁵.

On the Upper Half-Plane

For our use case, we are working with holomorphic functions on the upper half-plane. In order to transfer the results from the unit disk to the upper half-plane, consider the Cayley transform

The Cayley transform is a biholomorphic map with inverse given by

By pulling back the Poincaré metric from the disk via \(\phi^{-1}\), we obtain the Poincaré metric on the upper half-plane. For \(z_1, z_2 \in \mathbb{H}\), we have

After some algebraic manipulation, we can show that

Using the relationship between the Poincaré metric on the disk and the upper half-plane, we can transfer the results from the unit disk to the upper half-plane. We collect these results here.

Note that if \(g\) is a Möbius automorphism of the unit disk, then the composition \(f(z) = \phi(g(\phi^{-1}(z)))\) is a Möbius automorphism of the upper half-plane and takes exactly the same form as above. Furthermore, we may obtain the differential form of the Schwarz–Pick lemma for a holomorphic function \(f\) on the upper half-plane by defining \(\phi\circ f\circ \phi^{-1}\) and using the chain rule as well as the infinitesimal version of the Schwarz–Pick lemma on the disk.

Fixed-Point Theorems

In our random matrix example, for a fixed spectral parameter \(z \in \mathbb{H}\), the solution is defined in terms of \(m(z) \in \mathbb{H}\) which in turn is defined as the unique solution of a fixed-point equation

or, equivalently, as the solution of the (iterable) fixed-point map

For a fixed \(z \in \mathbb{H}\), one may check that the function \(G\) is a holomorphic map from the upper half-plane \(\mathbb{H}\) into itself.

There exists multiple fixed point theorems that could be used to show the existence of a unique solution \(m(z) \in \mathbb{H}\) for the fixed-point equation \(F(m;z)=1\) under various conditions. For instance, I like the Earle-Harris fixed point theorem⁶, which states that if \(f\) is a holomorphic map from a domain \(D\) in a complex Banach space into itself such that

1. \(f(D)\) is bounded in norm;
2. \(f\) is a strict contraction (i.e., there exists a constant \(\epsilon >0\) such that for all \(x \in D\) and \(y\notin D\), we have \(\|f(x) - y\| \geq \epsilon\));

then there exists a unique fixed point in \(D\). The proof of this theorem is based on the fact that such maps are strict contractions with respect to the CRF pseudometric (with uniform contraction constant) and hence we can apply the Banach fixed-point theorem.

The fixed-point theorem by Khatskevich-Reich-Shoikhe is also noteworthy ⁷. It states that we do not necessarily need the map to be a strict, and that uniformly continuous extensions on the boundary of the domain and strict contraction on the boundary are sufficient to guarantee the existence of a unique fixed point.

Here, I will instead present the Denjoy–Wolff theorem, a classical result in complex analysis that helps us understand the behavior of iterates of such maps. This theorem provides the basis for why \(m(z)\) is well-defined and unique within \(\mathbb{H}\).

Denjoy–Wolff Theorem

First, we state the theorem for the unit disk \(\mathbb{D}\), for which standard references are Milnor ⁸ and Carleson & Gamelin ⁹.

The limit point \(w_0\) is called the Denjoy–Wolff point of \(g\). We stated the proposition in a slightly different form than the standard one, basically using the fact that \(g\) cannot be an automorphism of the disk since \(g(\mathbb{D})\) is a proper subset of \(\mathbb{D}\).

Using the Cayley transform, we can state the corresponding result for the upper half-plane.

In the context of our fixed-point map \(G(m;z)\), we hope to apply the Denjoy–Wolff theorem to show that the iterates \(G^{\circ n}(m)\) converge to a unique point \(m(z) \in \mathbb{H}\) for each fixed \(z \in \mathbb{H}\). To do so, we need to check that the map \(G(m;z)\) is a proper subset of \(\mathbb{H}\), and then argue that the iterates do not converge to the boundary \(\partial\mathbb{H}\). The general strategy also extends to more general objects, such as matrix valued Herglotz functions.

Complex Implicit Function Theorem

The previous results give tools to show that the fixed-point equation \(F(m;z)=-1\) or \(G(m;z)=m\) has a unique solution in the upper half-plane for each fixed \(z \in \mathbb{H}\). However, we would like to know properties of the solution \(m(z)\) as a function of the spectral parameter \(z\). In particular, we want to show that the solution is holomorphic in the upper half-plane. To do so, we can use the complex implicit function theorem.

The proof follows from the holomorphic inverse-function theorem, and we do not discuss it here.

In our case, we want to apply the analytic implicit function theorem to the fixed-point equation \(F(m;z)=-1\). Taking the derivative with respect to \(m\), we have

By the differential form of the Schwarz–Pick lemma on the upper half-plane,

with strict inequality if \(G\) is not a Möbius automorphism of the upper half-plane, which is the case here. Hence, for \(z_0 \in \mathbb{H}\) fixed, if \(m_0\) is a solution of the fixed-point equation \(G(m_0;z_0)=m_0\), then

and consequently \(\partial_m F(m;z_0)\neq 0\) in a sufficiently small neighbourhood of \(m_0\). This means, by the complex implicit function theorem, that there exists a unique holomorphic function \(m(z)\) on a neighbourhood of \(z_0\) such that \(F(m(z);z)=0\). The argument we just presented is adapted from a paper by Elliot Paquette, Courtney Paquette, Lechao Xiao and Jeffrey Pennington ¹⁰.

Since the upper half-plane is simply connected and there is a unique solution for every fixed \(z \in \mathbb{H}\), we can glue the local branches together to obtain a global holomorphic function \(m : \mathbb{H} \to \mathbb{H}\) such that \(F(m(z);z)=0\) for all \(z \in \mathbb{H}\).

Schwarz Reflection Principle

The above gives us an analytic solution in the upper half-plane. However, in some cases, we may want to extend the solution to the lower half-plane and on part of the real line. This is important, for example, when we want to take contour integrals around some of the eigenvalues of the random matrix. This is where the Schwarz reflection principle comes into play.

The Schwarz reflection principle states that if \(f\) is a holomorphic function on the upper half-plane \(\mathbb{H}\), and if \(f\) extends continuously to an open interval \(I \subseteq \mathbb{R}\) with \(f(I) \subseteq \mathbb{R}\), then we can extend \(f\) to the lower half-plane by reflection. In other words, the function

is a holomorphic function on \(\mathbb{H} \cup I \cup -\mathbb{H}\), where we defined by \(f(x)=\lim_{t \to 0^+} f(x+it)\) for \(x \in I\). This follows from the Schwarz reflection principle.

For Herglotz functions, which are ubiquitous in random matrix theory, the non-tangential limits \(\lim_{t \to 0^+} f(x+it)\) exist for almost every \(x \in \mathbb{R}\). When dealing with Stieljes transform of empirical spectral measures, the non-tangential limit is related to the density of the associated measure. In this case, real tangential limits in an open interval \(I\) indicate that the spectral measure is supported away from \(I\).

References