Branching Brownian motion versus Random Energy Model in the supercritical phase: overlap distribution and temperature susceptibility

Abstract

In comparison with Derrida's REM, we investigate the influence of the so-called decoration processes arising in the limiting extremal processes of numerous log-correlated Gaussian fields. In particular, we focus on the branching Brownian motion and two specific quantities from statistical physics in the vicinity of the critical temperature. The first one is the two-temperature overlap, whose behavior at criticality is smoothened by the decoration process - unlike the one-temperature overlap which is identical - and the second one is the temperature susceptibility, as introduced by Sales and Bouchaud, which is strictly larger in the presence of decorations and diverges, close to the critical temperature, at the same speed as for the REM but with a different multiplicative constant. We also study some general decorated cases in order to highlight the fact that the BBM has a critical behavior in some sense to be made precise.

arXiv comment53 pages. Version accepted for publication in J Stat Phys

References in corpus (15)

• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
• Extreme local extrema of two-dimensional discrete Gaussian free field
• Extremes of the discrete two-dimensional Gaussian free field
• Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field
• Freezing and decorated Poisson point processes
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field
• 1-stable fluctuations in branching Brownian motion at critical temperature I: the derivative martingale
• Rejuvenation in the Random Energy Model
• Particle-number distribution in large fluctuations at the tip of branching random walks
• Fisher-KPP equation with small data and the extremal process of branching Brownian motion
• One step replica symmetry breaking and overlaps between two temperatures
• Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion