Modeling of bumping routes in the RSK algorithm and analysis of their approach to limit shapes

Introduction: The RSK algorithm establishes an equivalence of finite sequences of elements of linearly ordered sets and pairs of Young tableaux P and Q of the same shape. Of particular interest is the study of the asymptotic limit, i. e., the limit shape of the so-called bumping routes formed by the boxes of tableau P affected in a single iteration of the RSK algorithm. The exact formulae for these limit shapes were obtained by D. Romik and P. Śniady in 2016. However, the problem of investigating the dynamics of the approach of bumping routes to their limit shapes remains insufficiently studied. Purpose: To study the dynamics of distances between the bumping routes and their limit shapes in Young tableaux with the help of computer experiments. Results: We have obtained a large number of experimental bumping routes through a series of computer experiments for Young tableaux P of sizes up to 4·106, filled with real numbers in the range [0, 1] and sets of inserted values α Î [0.1, 0.15, … , 0.85]. We have compared these bumping routes in the L2 metric with the corresponding limit shapes and have calculated the average distances and variances of their deviations from the limit shapes. We present an empirical formula for the rate of approach of discretized bumping routes to their limit shapes. Also, the experimental parameters of the normal distributions of the deviations of the bumping routes are obtained for various input values.