Branching Brownian motion (BBM) is a convenient representative of the class of log-correlated random fields. Motivated by the conjectured criticality of the log-correlated fields, we take the viewpoint of statistical physics on the BBM: We consider the partition function of the field of energies given by the “positions” of the particles of the complex-valued BBM. In such a complex BBM energy model, we allow for arbitrary correlations between the real and imaginary parts of the energies. We identify the fluctuations of the partition function. As a consequence, we get the full phase diagram of the log-partition function. It turns out that the phase diagram is the same as for the field of independent energies, i.e., Derrida’s random energy model (REM). Yet, the fluctuations are different from those of the REM in all phases. All results are shown for any correlation between the real and imaginary parts of the random energy.
"The phase diagram of the complex branching Brownian motion energy model." Electron. J. Probab. 23 1 - 27, 2018. https://doi.org/10.1214/18-EJP245