Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields

@article{Fyodorov2009StatisticalMO,
  title={Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields},
  author={Yan V. Fyodorov and Pierre Le Doussal and Alberto Rosso},
  journal={Journal of Statistical Mechanics: Theory and Experiment},
  year={2009},
  volume={2009},
  pages={P10005}
}
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 temperature. The random potential sequences represent various versions of the 1/f noise generated by sampling the two-dimensional Gaussian free field (2D GFF) along various planar curves. Our method extends the recent analysis of Fyodorov and Bouchaud (2008 J. Phys. A: Math. Theor. 41 372001) from the… 

Freezing transitions and extreme values: random matrix theory, ζ ( 12 + i t ) and disordered landscapes

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

Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions.

TLDR
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.

Freezing transitions and extreme values: random matrix theory, and disordered landscapes

  • Y. FyodorovJ. Keating
  • 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

Disordered statistical physics in low dimensions: extremes, glass transition, and localization

TLDR
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.

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

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

Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise

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

Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.

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

POISSON-DIRICHLET STATISTICS FOR THE EXTREMES OF A LOG-CORRELATED GAUSSIAN FIELD

TLDR
At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1, which is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable.

The distribution of Gaussian multiplicative chaos on the unit interval

We consider the sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure.
...

References

SHOWING 1-10 OF 53 REFERENCES

Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces

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.

Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models.

TLDR
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.

FAST TRACK COMMUNICATION: Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential

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

Finite-size scaling in extreme statistics.

TLDR
A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well.

Maximal height statistics for 1/f(alpha) signals.

TLDR
Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions, which are found to be in agreement with simulations.

Extremes of the discrete two-dimensional Gaussian free field

We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to

Entropic repulsion of the lattice free field

Consider the massless free field on thed-dimensional lattice ℤd,d≧3; that is the centered Gaussian field on with covariances given by the Green function of the simple random walk on ℤd. We show that

Mellin Transform of the Limit Lognormal Distribution

The technique of intermittency expansions is applied to derive an exact formal power series representation for the Mellin transform of the probability distribution of the limit lognormal multifractal

Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

The multifractal scaling exponents are calculated for the critical wave function of a two-dimensional Dirac fermion in the presence of a random magnetic field. It is shown that the problem of
...