Combinatorial encoding of Bernoulli schemes and the asymptotic behavior of Young tableaux

TL;DR

This paper explores two combinatorial encodings of Bernoulli schemes, analyzing their decodability and asymptotic behavior of Young tableaux, with implications for representation theory and limit shape phenomena.

Contribution

It demonstrates the decodability of the RSK-based encoding and studies the asymptotic shape of Bernoulli variables on P-tableaux, introducing new dynamics.

Findings

Decodability of RSK-based encoding established

Limit 3D-shape of Bernoulli variables on P-tableaux identified

Alternative proof using representation theory proposed

Abstract

We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson--Schensted--Knuth) correspondence. In the first case, the decodability is a quite simple fact, while in the second case, this is a nontrivial result obtained by D.~Romik and P.~\'Sniady and based on the papers~ \cite{KV}, \cite{VK}, and others. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur--Weyl duality, will be presented elsewhere. We also study a new dynamics of Bernoulli variables on $P$-tableaux and find the limit 3D-shape of these tableaux.