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 $W_n:=\{(x_1,\dots ,x_n)\in {ℝ}^n:x_1<\dots <x_n\}$, 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 $A\in {ℝ}^{n\times n}$ Hermitian matrices, with probability density function $\rho (A)\propto e^{-\frac{1}{2}tr(A^2)}$.

Consider a Hermitian matrix $A\in {ℝ}^{n\times n}$. The space of Hermitian matrices can be mapped to the space of real vectors ${ℝ}^{n^2}$: $$A\mapsto (A_{11},\dots ,A_{nn},\sqrt{2}Re(A_{12}),\dots ,\sqrt{2}Re(A_{n-1,n}),\sqrt{2}Im(A_{12}),\dots ,\sqrt{2}Im(A_{n-1,n}))$$This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in ${ℝ}^{n^2}$ 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$$d{\lambda }_i=d{B}_i+\sum _{1\le j\le n:j\ne i}\frac{dt}{{\lambda }_i-{\lambda }_j}$$Template:Hidden begin

Infinitesimal generator

Template:Main 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 $F:{ℝ}^n\to ℝ$ with bounded derivatives, by direct differentiation, we have the Kolmogorov backward equation ${\partial }_t𝔼[F]=𝔼[D^{*}F]$. 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 ${ℝ}^n$ 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

Harish-Chandra-Itzykson-Zuber integral formula

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

References

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