# One step replica symmetry breaking and extreme order statistics of logarithmic REMs

@article{Cao2016OneSR,
title={One step replica symmetry breaking and extreme order statistics of logarithmic REMs},
author={Xiangyu Cao and Yan V. Fyodorov and Pierre Le Doussal},
journal={arXiv: Statistical Mechanics},
year={2016}
}
• Published 7 October 2016
• Mathematics
• arXiv: Statistical Mechanics
Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy…

## Figures from this paper

Disordered statistical physics in low dimensions: extremes, glass transition, and localization
This thesis presents original results in two domains of disordered statistical physics: logarithmic correlated Random Energy Models (logREMs), and localization transitions in long-range random matrices, and argues that such localization transitions occur generically in the broadly distributed class.
Finite Size Corrections to the Parisi Overlap Function in the GREM
• Mathematics
• 2017
We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model in two situations where replica symmetry is broken in the thermodynamic
Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions.
• Physics
Physical review. E
• 2018
It is argued that a seemingly nonphysical vanishing of the moment generating function for some values of parameters is related to the termination point transition (i.e., prefreezing), and the associated universal log corrections in the frozen phase are studied, both for logREMs and for the standard REM.
Extremes of log-correlated random fields and the Riemann zeta function, and some asymptotic results for various estimators in statistics
In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction
A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications
Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry–Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic
Statistics of Extremes in Eigenvalue-Counting Staircases.
• Mathematics
Physical review letters
• 2020
An extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields is used to obtain the cumulants of the distribution of that maximum for any β>0, and the results are expected to apply to the statistics of zeroes of the Riemann Zeta function.
A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures
A theory of intermittency differentiation is developed for a general class of 1D infinitely divisible multiplicative chaos measures. The intermittency invariance of the underlying infinitely
Physique statistique des systèmes désordonnées en basses dimensions
Cette these presente des resultats nouveaux dans deux sujets de la physique statistique du desordre: les modeles aux energies aleatoires logarithmiquement correlees (logREMs), et la transition de
Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance
• Frédéric Ouimet
• Mathematics
Latin American Journal of Probability and Mathematical Statistics
• 2017
We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V_N and adapt a technique of Bovier and
A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMC
• D. Ostrovsky
• Mathematics
Probability and Mathematical Physics
• 2021
A three parameter family of probability distributions is constructed such that its Mellin transform is defined over the same domain as the 2D GMC on the Riemann sphere with three insertion points

## References

SHOWING 1-10 OF 65 REFERENCES
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
Freezing transitions and extreme values: random matrix theory, and disordered landscapes
• Mathematics
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• 2014
We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution
Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces
• Mathematics
• 2008
We construct an N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment.
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
Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together
• Physics
• 2008
We compute the Functional Renormalization Group (FRG) disorder-correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space
On the distribution of the maximum value of the characteristic polynomial of GUE random matrices
• Mathematics
• 2016
Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random N×N matrices H from the Gaussian unitary ensemble
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
Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.
• Mathematics
Physical review letters
• 2012
We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the
Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise
• Mathematics
• 2012
Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of