Timeline for Is this Markov chain of Gaussian matrix products $G_1 G_2 \dots G_m$ ergodic?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2023 at 19:02 | comment | added | kvphxga | I'm not terribly familiar, but I definately have some grasp of Lyapunov exponents | |
| Jan 5, 2023 at 16:16 | comment | added | R W | Are you familiar with the Lyapunov exponents? It is in their terms that the boundary behaviour of random matrix products is described. | |
| Jan 5, 2023 at 15:06 | comment | added | kvphxga | I would just allow myself one follow-up. Is the $n$-sphere boundary of some hyperbolic model? Namely, if we only take the left eigenvectors of the Gaussian matrices and build a chain of them, corresponding to a random walk on the $n$-sphere, would that be ergodic? | |
| Jan 5, 2023 at 15:02 | comment | added | kvphxga | these all sound very fascinating, but I'm not able to fully understand the ideas. Since I don't want to bother with tens of follow up questions, Is there some reading material on these subjects that is accessible to non-mathematicians? (ie, no prior knowledge of differential geometry, hyperbolic spaces, etc) | |
| Jan 5, 2023 at 13:43 | comment | added | R W | The right way is, instead of a normalization, to pass to the random walk on an appropriate homogeneous space of the linear group, namely, on the flag space. For $SL(2,\mathbb R)$ this is the boundary of the hyperbolic plane, and in the upper half plane model the corresponding action is the action of $SL(2,\mathbb R)$ on the extended real line by linear fractional transformations. The resulting Markov chain is then very much ergodic. | |
| Jan 5, 2023 at 13:05 | comment | added | kvphxga | Is there some kind of normalization on $G_i$s that can make the chain ergodic? Namely, if we normalize by $\ell^1$ or $\ell^2$ or product of singular values of $G_i$s, will the chain become ergodic? | |
| Jan 5, 2023 at 11:59 | vote | accept | kvphxga | ||
| Jan 5, 2023 at 1:28 | history | answered | R W | CC BY-SA 4.0 |