Randomized Schützenberger's jeu de taquin and approximate calculation of co-transition probabilities of a central Markov process on the 3D Young graph

There exists a well-known hook-length formula for calculating the dimensions of 2D Young diagrams. Unfortunately, the analogous formula for 3D case is unknown. We introduce an approach for calculating the estimations of dimensions of three-dimensional Young diagrams also known as plane partitions. The most difficult part of this task is the calculation of co-transition probabilities for a central Markov process. We propose an algorithm for approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In case the generated paths are uniformly distributed, the proportion of paths passing through a certain branch gives us an approximate value of the co-transition probability. As our numerical experiments show, the random generator based on the randomized variant of the Schützenberger transformation allows to obtain accurate values of co-transition probabilities. Also a method to construct 3D Young diagrams with large dimensions is proposed.