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

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.

Deducing a Variational Principle with Minimal A Priori Assumptions

We study the well-known variational and large deviation principle for graph homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method to deduce those principles under minimal a priori assumptions. The only ingredient specific to the model is a discrete Kirszbraun theorem i.e. an extension theorem for graph homomorphisms. All other ingredients are of a general nature not specific to the model. They include elementary combinatorics, the compactness of Lipschitz functions, and a simplicial Rademacher theorem. Compared to the literature, our proof does not need any other preliminary results like e.g. concentration or strict convexity of the local surface tension. Therefore, the method is very robust and extends to more complex and subtle models, as e.g. the homogenization of limit shapes or graph-homomorphisms to a regular tree.

Grove Arctic Curves from Periodic Cluster Modular Transformations

Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of ergodic Gibbs measures in the resistor network model.