Mixing times of Markov chains for self-organizing lists and biased permutations

We study the mixing time of a Markov chain on biased permutations, a problem related to self-organizing lists. We are given probabilities {pi,j},$$ \left\{{p}_{i,j}\right\}, $$ for all i not equal j,$$ i\ne j, $$ such that pi,j=1-pj,i$$ {p}_{i,j}=1-{p}_{j,i} $$. The chain Mnn$$ {\mathcal{M}}_{nn} $$ iteratively chooses two adjacent elements i$$ i $$ and j$$ j $$, and swaps them with probability pi,j$$ {p}_{i,j} $$. It has been conjectured that Mnn$$ {\mathcal{M}}_{nn} $$ is rapidly mixing whenever the set of probabilities are "positively biased," that is, {pi,j >= 1/2},$$ \left\{{p}_{i,j}\ge 1/2\right\}, $$ for all i更多