Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$'s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$, and $X_0$ is some deterministic matrix with full rank fixed scale, $\|X_0\|_F^2=n$ and $\operatorname{rank}(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $\operatorname{Pr}(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$'s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.

Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$'s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$, and $X_0$ is some deterministic matrix with full rank fixed scale, $\|X_0\|_F^2=n$ and $\operatorname{rank}(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling(*) of two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $P(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$'s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.

(*) More elaborately, can we design $\{(G_t,W_t)\}_{t\in N}$ where both $G_1,\dots, $ and $W_1,\dots$ are independent, both jointly are coupled to lead to mixing of the two chains?

Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$ and $X_0$ is some deterministic matrix with full rank fixed scale $\|X_0\|_F^2=n, rank(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $Pr(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.

Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$'s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$, and $X_0$ is some deterministic matrix with full rank fixed scale, $\|X_0\|_F^2=n$ and $\operatorname{rank}(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $\operatorname{Pr}(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$'s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.

Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1)^{n\times n}$ and $X_0$ is some deterministic matrix with full rank fixed scale $\|X_0\|_F^2=n, rank(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $Pr(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.

Consider Markov chain $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ defined by $X_{t} = X_0 G_1 \dots G_t$ where $G_i$s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1/n)^{n\times n}$ and $X_0$ is some deterministic matrix with full rank fixed scale $\|X_0\|_F^2=n, rank(X_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X_0$ and $Y_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $Pr(X_t\neq Y_t) \le \alpha^t$ for some $\alpha<1$?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.