Randomized Schützenberger’s Jeu De Taquin and Approximate Calculation of the Cotransition Probabilities of a Central Markov Process on the 3D Young Graph

Abstract

There exists a well-known hook-length formula for calculating the dimensions of two-dimensional Young diagrams. Unfortunately, an analogous formula for the three-dimensional case is unknown. We introduce an approach to calculating estimated dimensions of three-dimensional Young diagrams, also known as plane partitions. The most difficult part of this task is the calculation of the cotransition probabilities for a central Markov process. We suggest an algorithm for an approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In the case where the generated paths are uniformly distributed, the proportion of paths passing through a certain edge gives us an approximate value of the cotransition probability. As our numerical experiments show, a random generator based on the randomized version of the Schutzenberger transformation allows one to obtain accurate values of cotransition probabilities. Also, a method for constructing three-dimensional Young diagrams with large dimensions is suggested. Bibliography: 13 titles.