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2018 The phase diagram of the complex branching Brownian motion energy model
Lisa Hartung, Anton Klimovsky
Electron. J. Probab. 23: 1-27 (2018). DOI: 10.1214/18-EJP245


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.


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Lisa Hartung. Anton Klimovsky. "The phase diagram of the complex branching Brownian motion energy model." Electron. J. Probab. 23 1 - 27, 2018.


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

zbMATH: 07021683
MathSciNet: MR3896864
Digital Object Identifier: 10.1214/18-EJP245

Primary: 60F05 , 60G70 , 60J80 , 60K35 , 82B44

Keywords: Branching Brownian motion , central limit theorem , Gaussian processes , logarithmic correlations , martingale convergence , phase diagram , random energy model , random variance


Vol.23 • 2018
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