# The phase diagram of the complex branching Brownian motion energy model

## TL;DR

This paper completes the phase diagram analysis of the complex branching Brownian motion energy model, establishing limit theorems and convergence properties across all phases and boundaries for various correlations.

## Contribution

It provides a comprehensive analysis of the phase diagram, including new limit theorems and convergence results for the model's partition function across all phases and boundaries.

## Key findings

- Proves a central limit theorem with random variance in Phase III and boundary regions.
- Establishes almost sure and L^1 martingale convergence in Phase I and boundary I/II.
- Analyzes the effects of correlation between real and imaginary parts of the energy.

## Abstract

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy.