# 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…

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## 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

- 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.…

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

- PhysicsPhysical 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.

### 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…

### Finite-size scaling in extreme statistics.

- Mathematics, PhysicsPhysical review letters
- 2008

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.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

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

- Mathematics
- 2006

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

- Mathematics, Physics
- 1995

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

- Mathematics
- 2009

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

- Physics
- 1997

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…