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The glassy phase of the complex branching Brownian motion energy model. (English) Zbl 1329.60303
Summary: We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called “glassy phase” – the regime of parameters where the behaviour of the partition function is governed by the extrema of the BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends a recent result of T. Madaule et al. [Commun. Math. Phys. 334, No. 3, 1157–1187 (2015; Zbl 1322.60177)], where the uncorrelated case was treated. In particular, our result covers the case of the real-valued BBM energy model at complex temperatures.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J65 Brownian motion
60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G15 Gaussian processes
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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