Asymptotic medians of random permutations sampled from reversal random walks

Medians can serve as a good estimator of the ancestor (the initial state) for k independent reversal random walks on the space of signed permutations before time n4, that is the identity permutation is a median of k random genomes sampled from k independent random walks at time cn where c≤1/4. In this paper we relax the time scale of the individual random walks, investigate the positions of all possible medians other than the initial state, and reduce the state space necessary for median search algorithms.