Comments on: 254A, Notes 3b: Brownian motion and Dyson Brownian motion

By: The Harish-Chandra-Itzykson-Zuber integral formula « What’s new

The Harish-Chandra-Itzykson-Zuber integral formula « What’s new — Sat, 09 Feb 2013 01:47:45 +0000

[...] motion (as well as the closely related formulae for the GUE ensemble), which were derived in this previous blog post. Both of these approaches can be found in several places in the literature (and I do not actually [...]

By: Some notes on Bakry-Emery theory « What’s new

Some notes on Bakry-Emery theory « What’s new — Tue, 05 Feb 2013 22:47:21 +0000

[...] is a stochastic process with initial probability distribution ; see for instance this previous blog post for more [...]

By: hera

hera — Mon, 04 Feb 2013 10:10:07 +0000

Great lecture! Prof. Tao, do you have ideas to simulate the Dyson’s brownian motion ? I’d like to do a computer simulation with MATLAB, can you give me some ideas?

By: A direct proof of the stationarity of the Dyson sine process under Dyson Brownian motion « What’s new

Sun, 11 Nov 2012 21:09:44 +0000

[...] of a Hermitian matrix under independent Brownian motion of its entries, and is discussed in this previous blog post. To cut a long story short, this stationarity tells us that the self-similar -point correlation [...]

By: alabair

alabair — Tue, 01 May 2012 18:52:23 +0000

I am amazed by the surprising number of knowledge that is difficult to me to digest. And by the way, I would like to hear about a white noise process. Good luck.

By: Terence Tao

Terence Tao — Thu, 19 Jan 2012 02:09:41 +0000

One can restore the confining potential by adding a term to the equation for , turning the Dyson Brownian motion to a Dyson Ornstein-Uhlenbeck process. This has the effect of keeping the variance of the matrix entries in the process constant, instead of growing linearly in time as is done here.

The two processes (normalised variance and non-normalised variance) can be easily rescaled to each other, so the choice of which one to use is basically a matter of taste.

By: Anonymous

Anonymous — Wed, 18 Jan 2012 20:12:21 +0000

Hi prof. Tau,

Thanks for these great notes!

Maybe this question is much too late, but I’d be very grateful for a reply. I’m used to think about the eigenvalues of the GUE as being subjected to a quadratic confining potential and mutual log repulsion: is there an easy way to understand the absence of the effect of the confining potential (which would give rise to a simple harmonic restoring force for each eigenvalue) in the formula for the Dyson Brownian motion, equation (5)?

Thanks for a great blog!

By: Diffusion in Ehrenfest wind-tree model « Disquisitiones Mathematicae

Diffusion in Ehrenfest wind-tree model « Disquisitiones Mathematicae — Fri, 18 Nov 2011 10:38:03 +0000

[...] the “justification” of the word “abnormal” comes by comparison with Brownian motion and/or central limit theorem: once we know that the diffusion is “sublinear” (maybe [...]

By: On Understanding Probability Puzzles | Nair Research Notes

On Understanding Probability Puzzles | Nair Research Notes — Sun, 21 Aug 2011 17:36:23 +0000

[...] by , from an experimental point of view. The formal construction for the Brownian motion can be found here for example. And you will get a good historical perspective of the Brownian motion here. One of the [...]

By: Dyson Brownian Motion | Research Notebook

Dyson Brownian Motion | Research Notebook — Fri, 15 Jul 2011 17:59:28 +0000

[...] main source for the material in this post is Terry Tao‘s set of lecture notes on Random Matrix Theory, though I also used Mehta (2004) and Anderson, Guinnet and Zeitouni (2009) as [...]