# Effect of microscopic pausing time distributions on the dynamical limit shapes for random Young diagrams

## TL;DR

This paper investigates how different microscopic pausing time distributions affect the large-scale limit shapes of Young diagrams, revealing both diffusive and anomalous behaviors influenced by heavy-tailed distributions.

## Contribution

It introduces a generalized continuous-time random walk on Young diagrams with arbitrary pausing time distributions and analyzes their impact on limit shape evolution.

## Key findings

- Diffusive scaling leads to a predictable limit shape evolution.
- Heavy-tailed pausing times cause anomalous, non-diffusive limit behaviors.
- Free probability theory effectively describes the time evolution of the limit shape.

## Abstract

The irreducible decomposition of successive restriction and induction of irreducible representations of a symmetric group gives rise to a Markov chain on Young diagrams keeping the Plancherel measure invariant. Starting from this Res-Ind chain, we introduce a not necessarily Markovian continuous time random walk on Young diagrams by considering a general pausing time distribution between jumps according to the transition probability of the Res-Ind chain. We show that, under appropriate assumptions for the pausing time distribution, a diffusive scaling limit brings us concentration at a certain limit shape depending on macroscopic time which leads to a similar consequence to the exponentially distributed case studied in our earlier work. The time evolution of the limit shape is well described by using free probability theory. On the other hand, we illustrate an anomalous phenomenon observed with a pausing time obeying a one-sided stable distribution, heavy-tailed without the mean, in which a nontrivial behavior appears under a non-diffusive regime of the scaling limit.