Dyson Brownian motion

In mathematics, the Dyson Brownian motion is a real-valued continuous-time stochastic process named for Freeman Dyson.^[1] Dyson studied this process in the context of random matrix theory.

There are several equivalent definitions:^[2]^[3]

Definition by stochastic differential equation: $d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}$ where $B_{1},...,B_{n}$ are different and independent Wiener processes. Start with a Hermitian matrix with eigenvalues ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ , 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 ${\textstyle n}$ independent Wiener processes started at different locations ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ , then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ .^[4]

Random matrix theory

Consider a Hermitian matrix ${\textstyle A\in \mathbb {R} ^{n\times n}}$ . The space of Hermitian matrices can be mapped to the space of real vectors ${\textstyle \mathbb {R} ^{n^{2}}}$ : $A\mapsto (A_{11},\dots ,A_{nn},{\sqrt {2}}Re(A_{12}),\dots ,{\sqrt {2}}Re(A_{n-1,n}),{\sqrt {2}}Im(A_{12}),\dots ,{\sqrt {2}}Im(A_{n-1,n}))$ This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in ${\textstyle \mathbb {R} ^{n^{2}}}$ maps back to a Brownian motion in the space of ${\textstyle n\times n}$ Hermitian matrices: $dA={\begin{bmatrix}dB_{11}&{\frac {1}{\sqrt {2}}}(dB_{12}+idB_{12}')&{\frac {1}{\sqrt {2}}}(dB_{13}+idB_{13}')&\cdots &{\frac {1}{\sqrt {2}}}(dB_{1n}+idB_{1n}')\\{\frac {1}{\sqrt {2}}}(dB_{12}-idB_{12}')&dB_{22}&{\frac {1}{\sqrt {2}}}(dB_{23}+idB_{23}')&\cdots &{\frac {1}{\sqrt {2}}}(dB_{2n}+idB_{2n}')\\{\frac {1}{\sqrt {2}}}(dB_{13}-idB_{13}')&{\frac {1}{\sqrt {2}}}(dB_{23}-idB_{23}')&dB_{33}&\cdots &{\frac {1}{\sqrt {2}}}(dB_{3n}+idB_{3n}')\\\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {1}{\sqrt {2}}}(dB_{1n}-idB_{1n}')&{\frac {1}{\sqrt {2}}}(dB_{2n}-idB_{2n}')&{\frac {1}{\sqrt {2}}}(dB_{3n}-idB_{3n}')&\cdots &dB_{nn}\end{bmatrix}}$ The claim is that the eigenvalues of $A$ evolve according to^[3] $d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}$

Proof

Proof

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

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

Since ${\textstyle G}$ is sampled from GUE(n), it is rotationally symmetric. Also, by assumption, the eigenvector ${\textstyle u_{i}}$ has norm 1, so ${\textstyle {\sqrt {dt}}u_{i}^{*}Gu_{i}}$ has the same distribution as ${\textstyle {\sqrt {dt}}e_{1}^{*}Ge_{1}}$ , which is distributed as ${\textstyle dB}$ .

Similarly, ${\textstyle \mathbb {E} [|u_{i}^{*}Gu_{j}|^{2}]=1}$ .

Infinitesimal generator

Define the adjoint Dyson operator: $D^{*}F:={\frac {1}{2}}\sum _{i=1}^{n}\partial _{\lambda _{i}}^{2}F+\sum _{1\leq i,j\leq n:i\neq j}{\frac {\partial _{\lambda _{i}}F}{\lambda _{i}-\lambda _{j}}}.$ 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 $D\rho :={\frac {1}{2}}\sum _{i=1}^{n}\partial _{\lambda _{i}}^{2}\rho -\sum _{1\leq i,j\leq n:i\neq j}\partial _{\lambda _{i}}\left({\frac {\rho }{\lambda _{i}-\lambda _{j}}}\right)$ Let $\rho (t,\lambda )=\Delta _{n}(\lambda )u(t,\lambda )$ , where $\Delta _{n}:=\prod _{i<j}(\lambda _{i}-\lambda _{j})$ 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 $\partial _{t}u={\frac {1}{2}}\sum _{i}\partial _{i}^{2}u$ ,

This can be proven by starting with the identity $\partial _{\lambda _{i}}\Delta _{n}=\Delta _{n}\sum _{1\leq j\leq n:i\neq j}{\frac {1}{\lambda _{i}-\lambda _{j}}}$ , then apply the fact that the Vandermonde determinant is harmonic: $\sum _{i}\partial _{i}^{2}\Delta _{n}=0$ .

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 $\mathrm {dim} (U(n))-\mathrm {dim} (U(2)\times U(1)^{n-2})=n^{2}-n-2$ 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 $u(0,\lambda )={\frac {1}{\Delta _{n}(\nu )}}\sum _{\sigma \in S_{n}}(-1)^{|\sigma |}\delta (\lambda -\sigma (\nu ))$ , so by the solution to the heat equation, and the Leibniz formula for determinants, we have^[5]

Johansson formula — Let ${\textstyle A_{0}}$ be a Hermitian matrix with simple spectrum ${\textstyle \nu =\left(\nu _{1},\ldots ,\nu _{n}\right)}$ , let ${\textstyle t>0}$ , and let ${\textstyle A_{t}=A_{0}+t^{1/2}G}$ where ${\textstyle G}$ is drawn from GUE. Then the spectrum ${\textstyle \lambda =\left(\lambda _{1},\ldots ,\lambda _{n}\right)}$ of ${\textstyle A_{t}}$ has probability density function

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

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.^[6]^[7]^[8]

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

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

Proof

Proof

Let the GUE(n) probability distribution over ${\textstyle V_{n}}$ be defined as ${\textstyle \rho (M)dM}$ , where ${\textstyle dM=dM_{11}\dots dM_{nn}d(Re(M_{12}))\dots d(Im(M_{n,n-1}))}$ , and ${\textstyle \rho (M)=C_{n}e^{-{\frac {1}{2}}tr(M^{2})}}$ and ${\textstyle C_{n}}$ is a constant. Similarly, the eigenvalue distribution for the GUE(n) is $\rho (\lambda )={\frac {1}{(2\pi )^{n/2}c_{n}}}\Delta _{n}(\lambda )^{2}e^{-\|\lambda \|_{2}^{2}/2}d\lambda$ where ${\textstyle d\lambda =d\lambda _{1}\dots d\lambda _{n}}$ , and ${\textstyle c_{n}}$ is another constant., and ${\textstyle \Delta _{n}=\prod _{1\leq i<j\leq n}\left(\lambda _{i}-\lambda _{j}\right)}$ is the Vandermonde determinant.

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

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

We compute ${\textstyle X}$ in one way.

Let ${\textstyle f_{t}(A):=f(A)e^{-{\frac {tr(A^{2})}{2t}}},g_{t}(B):=g(B)e^{-{\frac {tr(B^{2})}{2t}}}}$ , then the quantity is ${\begin{aligned}X&={\frac {C_{n}}{t^{n^{2}/2}}}\int _{V_{n}}\int _{V_{n}}f_{t}(A)e^{-{\frac {tr(AB)}{t}}}g_{t}(B)dAdB\\&={\frac {C_{n}}{t^{n^{2}/2}}}\int _{V_{n}}\int _{V_{n}}\int _{U(n)}f_{t}(A)e^{\frac {tr(AUBU^{*})}{t}}g_{t}(B)dAdBdU\\&={\frac {C_{n}}{t^{n^{2}/2}}}\int _{V_{n}}\int _{V_{n}}f_{t}(A)K_{t}(A,B)g_{t}(B)dAdB\\\end{aligned}}$ where we integrate over the Haar measure of the unitary group ${\textstyle U(n)}$ , and use the fact that ${\textstyle g_{t}(B)}$ is unitarily invariant, and we define the kernel
$K_{t}(A,B):=\int _{U(n)}e^{\frac {tr(AUBU^{*})}{t}}dU$

Since ${\textstyle K_{t},f_{t},g_{t}}$ are all unitarily invariant, we have $X={\frac {1}{C_{n}(2\pi )^{n}c_{n}^{2}t^{n^{2}/2}}}\int _{W_{n}}\int _{W_{n}}f_{t}(\lambda )g_{t}(\nu )K_{t}(\lambda ,\nu )\Delta _{n}(\lambda )^{2}\Delta _{n}(\nu )^{2}d\lambda d\nu$

We compute ${\textstyle X}$ in another way.

Fix ${\textstyle B}$ , then set ${\textstyle G=(A-B)/{\sqrt {t}}}$ , then we have $X=\int _{V_{n}}\int _{V_{n}}f(B+{\sqrt {t}}G)C_{n}e^{-{\frac {1}{2}}tr(G^{2})}dGdB$

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

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

References

^ Dyson, Freeman J. (1962-11-01). "A Brownian-Motion Model for the Eigenvalues of a Random Matrix". Journal of Mathematical Physics. 3 (6): 1191–1198. doi:10.1063/1.1703862. ISSN 0022-2488.
^ Bouchaud, Jean-Philippe; Potters, Marc, eds. (2020), "Dyson Brownian Motion", A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists, Cambridge: Cambridge University Press, pp. 121–135, ISBN 978-1-108-48808-2, retrieved 2023-11-25
^ ^a ^b Tao, Terence (2010-01-19). "254A, Notes 3b: Brownian motion and Dyson Brownian motion". What's new. Retrieved 2023-11-25.
^ Grabiner, David J. (1999). "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices". Annales de l'I.H.P. Probabilités et statistiques. 35 (2): 177–204. ISSN 1778-7017.
^ Johansson, Kurt Johansson (2001-01-01). "Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices". Communications in Mathematical Physics. 215 (3): 683–705. doi:10.1007/s002200000328. ISSN 1432-0916.
^ Harish-Chandra (1957). "Differential Operators on a Semisimple Lie Algebra". American Journal of Mathematics. 79 (1): 87–120. doi:10.2307/2372387. ISSN 0002-9327.
^ Itzykson, C.; Zuber, J.-B. (1980-03-01). "The planar approximation. II". Journal of Mathematical Physics. 21 (3): 411–421. doi:10.1063/1.524438. ISSN 0022-2488.
^ Tao, Terence (2013-02-09). "The Harish-Chandra-Itzykson-Zuber integral formula". What's new. Retrieved 2025-01-30.

[1] Dyson, Freeman J. (1962-11-01). "A Brownian-Motion Model for the Eigenvalues of a Random Matrix". Journal of Mathematical Physics. 3 (6): 1191–1198. doi:10.1063/1.1703862. ISSN 0022-2488.

[2] Bouchaud, Jean-Philippe; Potters, Marc, eds. (2020), "Dyson Brownian Motion", A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists, Cambridge: Cambridge University Press, pp. 121–135, ISBN 978-1-108-48808-2, retrieved 2023-11-25

[:0-3] Tao, Terence (2010-01-19). "254A, Notes 3b: Brownian motion and Dyson Brownian motion". What's new. Retrieved 2023-11-25.

[4] Grabiner, David J. (1999). "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices". Annales de l'I.H.P. Probabilités et statistiques. 35 (2): 177–204. ISSN 1778-7017.

[5] Johansson, Kurt Johansson (2001-01-01). "Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices". Communications in Mathematical Physics. 215 (3): 683–705. doi:10.1007/s002200000328. ISSN 1432-0916.

[6] Harish-Chandra (1957). "Differential Operators on a Semisimple Lie Algebra". American Journal of Mathematics. 79 (1): 87–120. doi:10.2307/2372387. ISSN 0002-9327.

[7] Itzykson, C.; Zuber, J.-B. (1980-03-01). "The planar approximation. II". Journal of Mathematical Physics. 21 (3): 411–421. doi:10.1063/1.524438. ISSN 0022-2488.

[8] Tao, Terence (2013-02-09). "The Harish-Chandra-Itzykson-Zuber integral formula". What's new. Retrieved 2025-01-30.

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