Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms

2004 ◽  
Vol 24 (3) ◽  
pp. 767-802 ◽  
Author(s):  
Y. GUIVARCH ◽  
A. N. STARKOV
Keyword(s):  
Random Walks ◽  
Linear Group ◽  
Homogeneous Spaces ◽  
Group Actions ◽  
Toral Automorphisms

scholarly journals The invariants of orthogonal group actions

1993 ◽  
Vol 48 (2) ◽  
pp. 313-319 ◽  
Author(s):  
Li Chiang ◽  
Yu-Ching Hung
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Orthogonal Group ◽  
Group Actions ◽  
Rational Function Field ◽  
Transcendental Extension

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

scholarly journals Limit profiles for reversible Markov chains

2021 ◽  
Author(s):  
Evita Nestoridi ◽  
Sam Olesker-Taylor
Keyword(s):  
Random Walk ◽  
Markov Chains ◽  
Random Walks ◽  
Gibbs Sampler ◽  
Statistical Physics ◽  
Homogeneous Spaces ◽  
New Method ◽  
Reversible Markov Chains ◽  
Similar Approximation ◽  
Random Transpositions

AbstractIn a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli  (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

Sharp ergodic theorems for group actions and strong ergodicity

1999 ◽  
Vol 19 (4) ◽  
pp. 1037-1061 ◽  
Author(s):  
ALEX FURMAN ◽  
YEHUDA SHALOM
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Spectral Condition ◽  
Ergodic Theorems ◽  
Averaging Operator ◽  
The Mean ◽  
Probability Measure Space

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

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