2012/04/10 by Ross G. Pinsky, Pinsky, Ross G.
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Genome Rearrangement Algorithms #Probability (math.PR) #Stochastic processes and statistical mechanics #math.CO #math.PR
paper · pdf · doi:10.48550/arxiv.1204.2081
arxiv created 2012/04/10 · openalex publication_date 2012/04/10 · arxiv updated 2012/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a permutation σ∈ Sn as a deck of cards numbered from 1 to n and laid out in a row, where σj denotes the number of the card that is in the j-th position from the left.\rm We define two cyclic to random transposition shuffles. The first one works as follows: for j=1,..., n, on the j-th step transpose the card that was \it originally\rm the j-th from the left with a random card (possibly itself). The second shuffle works as follows: on the j-th step, transpose the card that is \it currently\rm in the j-th position from the left with a random card (possibly itself). For these shuffles, for each b∈[0,1], we calculate explicitly the limiting rescaled density function of x,0≤ x≤1, for the probability that a card with a number around bn ends up in a position around xn, and for each x∈[0,1], we calculate the limiting rescaled density function of b,0≤ b≤ 1, for the probability that the card in a position around xn will be a card with a number around bn. These density functions all have a discontinuity at x=b, and for each of them, the supremum of the density is obtained by approaching the discontinuity from one side, and, for certain values of the parameter, the infimum of the density is obtained by approaching the discontinuity from the other side.