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Geometric Matrix Brownian Motion and the Lima Bean Law Kemp, Todd
Description
Geometric matrix Brownian motion is the solution (in $N\times N$ matrices) to the stochastic differential equation $dG_t = G_t dZ_t$, $G_0 = I$, where $Z_t$ is a Ginibre Brownian motion (all independent complex Brownian motion entries). It can also be described as the standard Brownian motion on the Lie group $\mathrm{GL}(N,\mathbb{C})$. For $N>2$, with probability $1$ it is not a normal matrix for any $t>0$. Over the last 5 years, we have made progress in understanding its asymptotic moments and fluctuations, but the non-normality (and lack of explicit symmetry) has made understanding its large-$N$ limit empirical eigenvalue distribution quite challenging. The tools around the circular law are now rich and provide a (log) potential course of action to understand the eigenvalues. There are two sides to this problem in general, both quite difficult: proving that the empirical law of eigenvalues converges (which amounts to certain tightness conditions on singular values), and computing what it converges {\em to}. In the case of the geometric matrix Brownian motion, the question of convergence is still a work in progress; but in recent joint work with Bruce Driver and Brian Hall, we have explicitly calculated the limit empirical eigenvalue distribution. It has an analytic density with a nice polar decomposition, supported on a region that resembles a lima bean for small $t>0$, then folds over and becomes a topological annulus when $t>4$. Our methods blend stochastic analysis, complex analysis, and PDE, and approach the log potential in a new way that we hope will be useful in a wider context.
Item Metadata
Title |
Geometric Matrix Brownian Motion and the Lima Bean Law
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-08-05T15:15
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Description |
Geometric matrix Brownian motion is the solution (in $N\times N$ matrices) to the stochastic differential equation $dG_t = G_t dZ_t$, $G_0 = I$, where $Z_t$ is a Ginibre Brownian motion (all independent complex Brownian motion entries). It can also be described as the standard Brownian motion on the Lie group $\mathrm{GL}(N,\mathbb{C})$. For $N>2$, with probability $1$ it is not a normal matrix for any $t>0$. Over the last 5 years, we have made progress in understanding its asymptotic moments and fluctuations, but the non-normality (and lack of explicit symmetry) has made understanding its large-$N$ limit empirical eigenvalue distribution quite challenging.
The tools around the circular law are now rich and provide a (log) potential course of action to understand the eigenvalues. There are two sides to this problem in general, both quite difficult: proving that the empirical law of eigenvalues converges (which amounts to certain tightness conditions on singular values), and computing what it converges {\em to}. In the case of the geometric matrix Brownian motion, the question of convergence is still a work in progress; but in recent joint work with Bruce Driver and Brian Hall, we have explicitly calculated the limit empirical eigenvalue distribution. It has an analytic density with a nice polar decomposition, supported on a region that resembles a lima bean for small $t>0$, then folds over and becomes a topological annulus when $t>4$.
Our methods blend stochastic analysis, complex analysis, and PDE, and approach the log potential in a new way that we hope will be useful in a wider context.
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Extent |
55.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: UC San Diego
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Series | |
Date Available |
2020-02-02
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0388519
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International