# Structure of the condensed phase in the inclusion process

## TL;DR

This paper provides a comprehensive analysis of condensation phenomena in the inclusion process, revealing hierarchical structures and phase behavior across various diffusion regimes using size-biased sampling and large deviation techniques.

## Contribution

It introduces a novel application of size-biased sampling to study the condensed phase structure in the inclusion process, including detailed large deviation results and hierarchical organization.

## Key findings

- Condensation can extend over multiple lattice sites.
- The condensed phase exhibits a hierarchical structure characterized by the Poisson-Dirichlet distribution.
- Results are supported by Monte-Carlo simulations and recursive numerics.

## Abstract

We establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion, covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number. We make use of size-biased sampling to study the structure of the condensed phase, which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the Poisson-Dirichlet distribution. While this approach is established in other areas including population genetics or random permutations, we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions. We discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase, and the connection to other generic particle systems. Our results are exact, and we present Monte-Carlo simulation data and recursive numerics for partition functions to illustrate the main points.