Kac's random walk

Workshop introduction

In general, we will use the following notation in the problem list. We consider aperiodic irreducible Markov chains on a state space, $\Omega$ with unique invariant (stationary) distribution $\pi$. In the discrete time setting we allow the matrix $(P)_{x,y\in\Omega}$ with $P_{xy}=P(x,y)$ to denote the transition matrix of the Markov chain.

A chain is reversible if $\pi(x)P(x,y)=\pi(y)P(y,x)$ for all $x,y\in\Omega$. $Q(x,y)=\pi(x)P(x,y)$ is the edge measure on $(x,y)$. We denote by $t_{\mbox{mix}}$ the total variation mixing time defined as

and set $t_{\mbox{mix}}=t_{\mbox{mix}}(\epsilon)$, where the total variation distance

Question context

Consider the Kac's random walk on the $n$-sphere defined as follows: for a vector $v\in \mathcal S^n$, pick two coordinates uniformly at random, and an angle uniformly at random on $[0,2\pi)$. Then rotate in the circle generated by the two coordinates by that angle.

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

For Kac's random walk, it is known that $\frac 12 n\log n \leq t_{\mbox{mix}} \leq 4n\log n$.

For the two coordinate random walk on $SO(n)$, $t_{\mbox{mix}}\leq O(n^4)$ but $\tilde O(n^2)$ is conjectured.