Electronic Journal of Probability

The phase diagram of the complex branching Brownian motion energy model

Lisa Hartung and Anton Klimovsky

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 127, 27 pp.

Received: 18 April 2017
Accepted: 16 November 2018
First available in Project Euclid: 19 December 2018

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Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Gaussian processes branching Brownian motion logarithmic correlations random energy model phase diagram central limit theorem random variance martingale convergence

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Hartung, Lisa; Klimovsky, Anton. The phase diagram of the complex branching Brownian motion energy model. Electron. J. Probab. 23 (2018), paper no. 127, 27 pp. doi:10.1214/18-EJP245. https://projecteuclid.org/euclid.ejp/1545188694

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