On the convergence of bumping routes to their limit shapes in the RSK algorithm: numerical experiments

Abstract

Introduction: The Robinson — Schensted — Knuth (RSK) algorithm transforms a sequence of elements of a linearly ordered set into a pair of Young tableaux P, Q of the same shape. This transformation is based on the process of bumping and inserting elements in tableau P according to special rules. The trajectory formed by all the boxes of the tableau P shifted in the RSK algorithm is called the bumping route. D. Romik and P. Śniady in 2016 obtained an explicit formula for the limit shape of the bumping route, which is determined by its first element. However, the rate of convergence of the bumping routes to the limit shape has not been previously investigated either theoretically or by numerical experiments. Purpose: Carrying out computer experiments to study the dynamics of the bumping routes produced by the RSK algorithm on Young tableaux as their sizes increase. Calculation of statistical means and variances of deviations of bumping routes from their limit shapes in the L2 metric for various values fed to the input of the RSK algorithm. Results: A series of computer experiments have been carried out on Young tableaux, consisting of up to 10 million boxes. We used 300 tableaux of each size. Different input values (0.1, 0.3, 0.5, 0.7, 0.9) were added to each such tableau using the RSK algorithm, and the deviations of the bumping routes built from these values from the corresponding limit shapes were calculated. The graphs of the statistical mean values and variances of these deviations were produced. It is noticed that the deviations decrease in proportion to the fourth root of the tableau size n. An approximation of the dependence of the mean values of deviations on n was obtained using the least squares method.

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

“…It was shown in [9] that the deviations of the bumping routes from the limit shapes for large values of n are well approximated by an empirically obtained function of the form…”

“…Thus, the convergence turns out to be quite slow. The results of calculating the deviations obtained in the framework of computer experiments were approximated using the function (9) with free parameters a and b. The table below shows the values of the obtained parameters for some input numbers , as well as the maximum deviations  of the approximation curves from the experimental bumping routes.…”

“…In this paper, we study the dynamics of convergence of the generated bumping routes to their limit shapes. Similar simulations have already been performed in [9], but compared to this work, the scheme of experiments was significantly changed. In particular, the number of Young tableaux involved in the simulation was increased, and a larger number of limit shapes of bumping routes, to which the experimental curves tend, was considered.…”

“…The geometry and combinatorics of the bumping routes of the RSK algorithm was studied [6][7][8][9][10] both theoretically and with the help of computer experiments. In [6], the asymptotic behaviour of the convergence of bumping routes to their limit shapes was studied, and exact equations describing such limit shapes were also obtained.…”

“…Since the end of the limit shape of a bumping route may not coincide with the end of the experimentally calculated bumping route, the projections of these functions onto the x axis are generally defined on different intervals. At the same time, in [9] to calculate the deviation of a bumping route from the theoretical limit shape, we scaled the experimental bumping route by multiplying its domain of definition by some constant so that its end coincides with the theoretical one. Finally, the L 2 distance between the scaled experimental bumping route and the theoretical curve was calculated.…”