Random biased transpositions

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

The random uniform transposition walk on $S_n$ is the discrete time Markov chain that at each time step applies a transposition $(i,j)$ uniformly at random.

The random nearly-uniform (biased) transposition walk is the corresponding chain that at each time step applies a transposition $(i,j)$ with probability $p_{ij}$ for $p_{ij}$ comparable to the uniform rate up to uniformly bounded constants.

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