Dyson Brownian motion

In mathematics, the Dyson Brownian motion is a real-valued continuous-time stochastic process named for Freeman Dyson. Dyson studied this process in the context of random matrix theory.

There are several equivalent definitions:

Definition by stochastic differential equation:$$d{\lambda }_i=d{B}_i+\sum _{1\le j\le n:j\ne i}\frac{dt}{{\lambda }_i-{\lambda }_j}$$where $B_1,...,B_n$ are different and independent Wiener processes. Start with a Hermitian matrix with eigenvalues ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$, then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion. This is defined within the Weyl chamber $W_n:=\{(x_1,\dots ,x_n)\in {\mathbb{ℝ}}^n:x_1<\dots <x_n\}$, as well as any coordinate-permutation of it.

Start with $n$ independent Wiener processes started at different locations ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$, then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$.

Random matrix theory

In Random Matrix Theory, the Gaussian unitary ensemble is a fundamental ensemble. It is defined as a probability distribution over the space of $A\in {\mathbb{ℝ}}^{n\times n}$ Hermitian matrices, with probability density function $\rho (A)\propto e^{-\frac{1}{2}tr(A^2)}$.

Consider a Hermitian matrix

$A\in {\mathbb{ℝ}}^{n\times n}$

. The space of Hermitian matrices can be mapped to the space of real vectors

${\mathbb{ℝ}}^{n^2}$

:

$$A\mapsto (A_{11},\dots ,A_{nn},\sqrt{2}Re(A_{12}),\dots ,\sqrt{2}Re(A_{n-1,n}),\sqrt{2}Im(A_{12}),\dots ,\sqrt{2}Im(A_{n-1,n}))$$

This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in

${\mathbb{ℝ}}^{n^2}$

maps back to a Brownian motion in the space of

$n\times n$

Hermitian matrices:

$$dA=\left[\begin{array}{ccccc}d{B}_{11}& \frac{1}{\sqrt{2}}(d{B}_{12}+id{B}_{1^{\prime }2})& \frac{1}{\sqrt{2}}(d{B}_{13}+id{B}_{1^{\prime }3})& \cdots & \frac{1}{\sqrt{2}}(d{B}_{1n}+id{B}_{1^{\prime }n})\\ \frac{1}{\sqrt{2}}(d{B}_{12}-id{B}_{1^{\prime }2})& d{B}_{22}& \frac{1}{\sqrt{2}}(d{B}_{23}+id{B}_{2^{\prime }3})& \cdots & \frac{1}{\sqrt{2}}(d{B}_{2n}+id{B}_{2^{\prime }n})\\ \frac{1}{\sqrt{2}}(d{B}_{13}-id{B}_{1^{\prime }3})& \frac{1}{\sqrt{2}}(d{B}_{23}-id{B}_{2^{\prime }3})& d{B}_{33}& \cdots & \frac{1}{\sqrt{2}}(d{B}_{3n}+id{B}_{3^{\prime }n})\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \frac{1}{\sqrt{2}}(d{B}_{1n}-id{B}_{1^{\prime }n})& \frac{1}{\sqrt{2}}(d{B}_{2n}-id{B}_{2^{\prime }n})& \frac{1}{\sqrt{2}}(d{B}_{3n}-id{B}_{3^{\prime }n})& \cdots & d{B}_{nn}\end{array}\right]$$

The claim is that the eigenvalues of

$A$

evolve according to

$$d{\lambda }_i=d{B}_i+\sum _{1\le j\le n:j\ne i}\frac{dt}{{\lambda }_i-{\lambda }_j}$$

Proof

Proof

Since each $dB$ is on the order of $O(\sqrt{dt})$, we can equivalently write $dA=\sqrt{dt}G$, where $G$ is a random Hermitian matrix where each entry is on the order of $O(1)$. By construction of the standard Brownian motion, $dA$ is independent of $A$, so $G$ is independent of $A$, and can be written as $$dA=\left[\begin{array}{ccccc}{X}_{11}& \frac{1}{\sqrt{2}}(X_{12}+i{Y}_{12})& \frac{1}{\sqrt{2}}(X_{13}+i{Y}_{13})& \cdots & \frac{1}{\sqrt{2}}(X_{1n}+i{Y}_{1n})\\ \frac{1}{\sqrt{2}}(X_{12}-i{Y}_{12})& X_{22}& \frac{1}{\sqrt{2}}(X_{23}+i{Y}_{23})& \cdots & \frac{1}{\sqrt{2}}(X_{2n}+i{Y}_{2n})\\ \frac{1}{\sqrt{2}}(X_{13}-i{Y}_{13})& \frac{1}{\sqrt{2}}(X_{23}-i{Y}_{23})& X_{33}& \cdots & \frac{1}{\sqrt{2}}(X_{3n}+i{Y}_{3n})\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \frac{1}{\sqrt{2}}(X_{1n}-i{Y}_{1n})& \frac{1}{\sqrt{2}}(X_{2n}-i{Y}_{2n})& \frac{1}{\sqrt{2}}(X_{3n}-i{Y}_{3n})& \cdots & X_{nn}\end{array}\right]$$ where each random variable $X_{ij},Y_{ij}$ is standard normal. In other words, $G$ is distributed according to the GUE(n).

By the first and second Hadamard variation formulas and Ito’s lemma, we have $${\lambda }_i(t+dt)={\lambda }_i(t)+\sqrt{dt}{u}_i^{*}G{u}_i+dt\sum _{j\ne i}\frac{\mathbb{𝔼}[|u_i^{*}G{u}_j{|}^2]}{{\lambda }_i-{\lambda }_j}$$

Since $G$ is sampled from GUE(n), it is rotationally symmetric. Also, by assumption, the eigenvector $u_i$ has norm 1, so $\sqrt{dt}{u}_i^{*}G{u}_i$ has the same distribution as $\sqrt{dt}{e}_1^{*}G{e}_1$, which is distributed as $dB$.

Similarly, $\mathbb{𝔼}[|u_i^{*}G{u}_j{|}^2]=1$.

Infinitesimal generator

Main article: Infinitesimal generator (stochastic processes)

Define the adjoint Dyson operator:$$D^{*}F:=\frac{1}{2}\sum _{i=1}^n\underset{{\lambda }_i}{\overset{2}{\partial }}F+\sum _{1\le i,j\le n:i\ne j}\frac{{\partial }_{{\lambda }_i}F}{{\lambda }_i-{\lambda }_j}.$$For any smooth function $F:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ with bounded derivatives, by direct differentiation, we have the Kolmogorov backward equation ${\partial }_t\mathbb{𝔼}[F]=\mathbb{𝔼}[D^{*}F]$. Therefore, the Kolmogorov forward equation for the eigenspectrum is $\partial \rho =D\rho$, where $D$ is the Dyson operator by$$D\rho :=\frac{1}{2}\sum _{i=1}^n\underset{{\lambda }_i}{\overset{2}{\partial }}\rho -\sum _{1\le i,j\le n:i\ne j}{\partial }_{{\lambda }_i}\left(\frac{\rho }{{\lambda }_i-{\lambda }_j}\right)$$Let $\rho (t,\lambda )={\mathrm{\Delta }}_n(\lambda )u(t,\lambda )$, where ${\mathrm{\Delta }}_n:=\prod _{i<j}({\lambda }_i-{\lambda }_j)$ is the Vandermonde determinant, then the time-evolution of eigenspectrum is equivalent to the time-evolution of $u$, which happens to satisfy the heat equation ${\partial }_{t}u=\frac{1}{2}\sum _i\underset{i}{\overset{2}{\partial }}u$,

This can be proven by starting with the identity ${\partial }_{{\lambda }_i}{\mathrm{\Delta }}_n={\mathrm{\Delta }}_n\sum _{1\le j\le n:i\ne j}\frac{1}{{\lambda }_i-{\lambda }_j}$, then apply the fact that the Vandermonde determinant is harmonic: $\sum _i\underset{i}{\overset{2}{\partial }}{\mathrm{\Delta }}_n=0$.

Johansson formula

Each Hermitian matrix with exactly two eigenvalues equal is stabilized by $U(2)\times U(1{)}^{n-2}$, so its orbit under the action of $U(n)$ has $\mathrm{d}\mathrm{i}\mathrm{m}(U(n))-\mathrm{d}\mathrm{i}\mathrm{m}(U(2)\times U(1{)}^{n-2})=n^2-n-2$ dimensions. Since the space of $n-1$ different eigenvalues is $(n-1)$-dimensional, the space of Hermitian matrix with exactly two eigenvalues equal has $n^2-3$ dimensions.

By a dimension-counting argument, $\rho$ vanishes at sufficiently high order on the border of the Weyl chamber, such that $u$ can be extended to all of ${\mathbb{ℝ}}^n$ by antisymmetry, and this extension still satisfies the heat equation.

Now, suppose the random matrix walk begins at some deterministic $A(0)$. Let its eigenspectrum be $\nu =\lambda (A(0))$, then we have $u(0,\lambda )=\frac{1}{{\mathrm{\Delta }}_n(\nu )}\sum _{\sigma \in S_n}(-1{)}^{|\sigma |}\delta (\lambda -\sigma (\nu ))$, so by the solution to the heat equation, and the Leibniz formula for determinants, we have

Johansson formula — Let $A_0$ be a Hermitian matrix with simple spectrum $\nu =({\nu }_1,\dots ,{\nu }_n)$, let $t>0$, and let $A_t=A_0+t^{1/2}G$ where $G$ is drawn from GUE. Then the spectrum $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ of $A_t$ has probability density function

$$\rho (t,\lambda )=\frac{1}{(2\pi t{)}^{n/2}}\frac{{\mathrm{\Delta }}_n(\lambda )}{{\mathrm{\Delta }}_n(\nu )}\det {\left(e^{-{({\lambda }_i-{\nu }_j)}^2/2t}\right)}_{1\le i,j\le n}$$

on the Weyl chamber.

Harish-Chandra-Itzykson-Zuber integral formula

Dyson Brownian motion allows a short proof of the Harish-Chandra-Itzykson-Zuber integral formula.

Harish-Chandra-Itzykson-Zuber integral formula — If $A,B$ have no repeated eigenvalues, and $t$ is a nonzero complex number, then - $$\underset{U(n)}{\int }\exp (t\mathrm{tr}\left(AUB{U}^{*}\right))dU=c_n\frac{\det {[\exp (t{\lambda }_i(A){\lambda }_j(B))]}_{1\le i,j\le n}}{t^{(n^2-n)/2}{\mathrm{\Delta }}_n(\lambda (A)){\mathrm{\Delta }}_n(\lambda (B))}$$

where $U$ is integrated over the Haar probability measure of the unitary group $U(n)$, and $c_n={\prod }_{i=1}^{n}i!$.

Proof

Proof

Let the GUE(n) probability distribution over $V_n$ be defined as $\rho (M)dM$, where $dM=d{M}_{11}\dots d{M}_{nn}d(Re(M_{12}))\dots d(Im(M_{n,n-1}))$, and $\rho (M)=C_n{e}^{-\frac{1}{2}tr(M^2)}$ and $C_n$ is a constant. Similarly, the eigenvalue distribution for the GUE(n) is $$\rho (\lambda )=\frac{1}{(2\pi {)}^{n/2}{c}_n}{\mathrm{\Delta }}_n(\lambda {)}^2{e}^{-\Vert \lambda {\Vert }_2^2/2}d\lambda$$ where $d\lambda =d{\lambda }_1\dots d{\lambda }_n$, and $c_n$ is another constant., and ${\mathrm{\Delta }}_n=\prod _{1\le i<j\le n}({\lambda }_i-{\lambda }_j)$ is the Vandermonde determinant.

If $f:V_n\to \mathbb{ℂ}$ is unitarily invariant, with sufficient regularity and decay, then it can be decomposed as $f(M_n)=f(\lambda (M_n))$. By Riesz representation theorem, there exists some function $w$ such that $\int f(M)dM=\int f(\lambda )w(\lambda )d\lambda$, which by the above argument equals $$w(\lambda )=\rho (\lambda )/\rho (M)=\frac{{\mathrm{\Delta }}_n(\lambda {)}^2}{c_n{C}_n(2\pi {)}^{n/2}}$$

Given two such unitarily invariant functions $f,g$ with sufficient regularity and decay, then consider their heat kernel convolution $$X=\iint dAdBf(A)\frac{C_n}{t^{n^2/2}}{e}^{-\frac{tr(A-B{)}^2}{2t}}g(B)$$

We compute $X$ in one way.

Let $f_t(A):=f(A)e^{-\frac{tr(A^2)}{2t}},g_t(B):=g(B)e^{-\frac{tr(B^2)}{2t}}$, then the quantity is $$\begin{array}{rl}X& =\frac{C_n}{t^{n^2/2}}\underset{V_n}{\int }\underset{V_n}{\int }{f}_t(A)e^{-\frac{tr(AB)}{t}}{g}_t(B)dAdB\\ & =\frac{C_n}{t^{n^2/2}}\underset{V_n}{\int }\underset{V_n}{\int }\underset{U(n)}{\int }{f}_t(A)e^{\frac{tr(AUB{U}^{*})}{t}}{g}_t(B)dAdBdU\\ & =\frac{C_n}{t^{n^2/2}}\underset{V_n}{\int }\underset{V_n}{\int }{f}_t(A)K_t(A,B)g_t(B)dAdB\\ \end{array}$$ where we integrate over the Haar measure of the unitary group $U(n)$, and use the fact that $g_t(B)$ is unitarily invariant, and we define the kernel
$$K_t(A,B):=\underset{U(n)}{\int }{e}^{\frac{tr(AUB{U}^{*})}{t}}dU$$

Since $K_t,f_t,g_t$ are all unitarily invariant, we have $$X=\frac{1}{C_n(2\pi {)}^n{c}_n^2{t}^{n^2/2}}\underset{W_n}{\int }\underset{W_n}{\int }{f}_t(\lambda )g_t(\nu )K_t(\lambda ,\nu ){\mathrm{\Delta }}_n(\lambda {)}^2{\mathrm{\Delta }}_n(\nu {)}^{2}d\lambda d\nu$$

We compute $X$ in another way.

Fix $B$, then set $G=(A-B)/\sqrt{t}$, then we have $$X=\underset{V_n}{\int }\underset{V_n}{\int }f(B+\sqrt{t}G)C_n{e}^{-\frac{1}{2}tr(G^2)}dGdB$$

Apply the Johansson formula, and convert the domain of integral to the Weyl chamber: $$X=\frac{1}{C_n{c}_n(2\pi {)}^n{t}^{n/2}}\iint f(\lambda )g(\nu ){\mathrm{\Delta }}_n(\lambda ){\mathrm{\Delta }}_n(\nu )\det {\left[e^{-\frac{({\lambda }_i-{\nu }_j{)}^2}{2t}}\right]}_{i,j}d\lambda d\nu$$

Equate the two results, and simplify, we obtain the equality.

Ginibre formula

Ginibre formula (Tao 2012, page 251) — $$\rho (\lambda )=\frac{1}{(2\pi {)}^{n/2}1!\dots n!}{e}^{-|\lambda {|}^2/2}{|{\mathrm{\Delta }}_n(\lambda )|}^2$$ on the Weyl chamber.

Proof

Proof

The GUE is constructed as the $t=1$ distribution when starting with $A_0$. So we take $t=1$ and $\nu \to 0$ in the Johansson formula.

$$\rho (\lambda )=\frac{{\mathrm{\Delta }}_n(\lambda )}{(2\pi {)}^{n/2}}\underset{\nu \to 0}{\lim }\frac{1}{{\mathrm{\Delta }}_n(\nu )}\det {\left(e^{-\frac{1}{2}{({\lambda }_i-{\nu }_j)}^2}\right)}_{1\le i,j\le n}$$

Since $e^{-{({\lambda }_i-{\nu }_j)}^2/2}=e^{-{\lambda }_i^2/2}{e}^{-{\nu }_j^2/2}{e}^{{\lambda }_i{\nu }_j}$, we have $$\det {\left(e^{-{({\lambda }_i-{\nu }_j)}^2/2}\right)}_{1\le i,j\le n}=e^{-|\lambda {|}^2/2}{e}^{-|\nu {|}^2/2}\det {\left(e^{{\lambda }_i{\nu }_j}\right)}_{1\le i,j\le n}$$

Now by a property of Vandermonde matrix, at the $\nu \to 0$ limit,

$$\det {\left(e^{{\lambda }_i{\nu }_j}\right)}_{1\le i,j\le n}=\frac{1}{1!\dots n!}{\mathrm{\Delta }}_n(\lambda ){\mathrm{\Delta }}_n(\nu )+o({\mathrm{\Delta }}_n(\nu ))$$ uniformly in $\nu$.

References

• Tao, Terence (2012). Topics in random matrix theory. Graduate studies in mathematics. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-7430-1. 
• Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30). "10. Addition of Large Random Matrices". A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists (1 ed.). Cambridge University Press. doi:10.1017/9781108768900. ISBN 978-1-108-76890-0. https://www.cambridge.org/core/product/identifier/9781108768900/type/book.