# Glassy phase and freezing of log-correlated Gaussian potentials

title={Glassy phase and freezing of log-correlated Gaussian potentials},
author={Thomas Madaule and R{\'e}mi Rhodes and Vincent Vargas},
journal={Annals of Applied Probability},
year={2013},
volume={26},
pages={643-690}
}
• Published 21 October 2013
• Mathematics
• Annals of Applied Probability
In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos constructed in \cite{Rnew7,Rnew12}. This could be seen as a first rigorous step in the…

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## References

SHOWING 1-10 OF 111 REFERENCES
Exact Asymptotics of the Freezing Transition of a Logarithmically Correlated Random Energy Model
We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a
FAST TRACK COMMUNICATION: Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential
• Mathematics
• 2008
We investigate some implications of the freezing scenario proposed by Carpentier and Le Doussal (CLD) for a random energy model (REM) with logarithmically correlated random potential. We introduce a
Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models.
• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2001
Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (corresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to recover the multifractal spectrum in the delocalized regime.
Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Physics
• 2000
We study via RG, numerics, exact bounds and qualitative arguments the equilibrium Gibbs measure of a particle in a $d$-dimensional gaussian random potential with {\it translationally invariant
Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields
• Mathematics
• 2009
We compute the distribution of the partition functions for a class of one-dimensional random energy models with logarithmically correlated random potential, above and at the glass transition
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• Mathematics
• 2012
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching
Polymers on disordered trees, spin glasses, and traveling waves
• Materials Science
• 1988
We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions
Complex Gaussian Multiplicative Chaos
• Mathematics
• 2013
In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all
Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field
• Mathematics
• 2013
In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the