Process \eqref{eq:20151125b} is a random process with mean version to zero because of the \( -\alpha \lambda_i dt \) term. The eigenvalues \( \lambda_i(t) \) also repel one another because of the term \( \sum_{ j \neq i} \dfrac{1}{\lambda_i - \lambda_j} \). I now illustrate this behaviour for \( \alpha = 20 \) and \( n = 5 \). The code is quite similar to the code I used in the previous blog post. The only difference is that I use the following code to simulate \( X_t \)

```
(*Simulates a path of the Dyson Ornstein-Uhlenbeck process
```

```
in:
The dimension of the matrices is size x size.
```

`T is the maximum time to simulate. `

`NSteps is in how many steps you simulate. `

```
out:
a list of matrices X_t*)
```

```
SimulateOnePathOU[alpha_, Size_, T_, NSteps_] :=
Module[{dt = T/NSteps, t, H, path},
```

`path = ConstantArray[0, NSteps + 1]; `

`path[[1]] = ConstantArray[0, {Size, Size}];`

`For[t = 1, t <= NSteps, t++, H = RandomMatrixGUE[Size]; `

`path[[t + 1]] = - alpha path[[t]] dt + path[[t]] + H Sqrt[dt]]; `

`path] `

This produces the following graph

The eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_5 \) as function of time |

Related: Illustration of Dyson Brownian motion

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