Multifractality and Freezing Phenomena in Random Energy Landscapes: an Introduction
arXiv:0911.2765 · doi:10.1016/j.physa.2010.01.020
Abstract
The Boltzmann-Gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet nontrivial example of disorder-induced multifractal measures. We introduce and discuss two analytically tractable models for such potentials. The first model uses multiplicative cascades and is equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by B. Derrida and H. Spohn. Second model is the infinite-dimensional version of the problem in Euclidean space which can be solved by employing the replica trick. In particular, the latter version allows one to identify the freezing of multifractality exponents with a mechanism of the replica symmetry breaking (RSB) and to elucidate its physical meaning. The corresponding 1-step RSB solution turns out to be {\it marginally stable} everywhere in the low-temperature phase. In the end we put the model in a more general context by relating to the Gaussian Free Field and briefly discussing recent developments and extensions. The appendices provide a detailed exposition of the replica analysis of the model discussed in the lectures.
Extended version of lecture notes for the International Summer School "Fundamental Problems in Statistical Physics XII" at Leuven, Belgium on Aug 31 - Sept 11, 2009 ; 34 pages
References in corpus (8)
- Anderson Transitions
- 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 Noises generated by Gaussian Free Fields
- Extremes of the discrete two-dimensional Gaussian free field
- Dynamics of trap models
- Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
- Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
- A causal multifractal stochastic equation and its statistical properties
Cited by in corpus (32)
- Extreme value statistics of correlated random variables: a pedagogical review
- Freezing Transitions and Extreme Values: Random Matrix Theory, , and Disordered Landscapes
- Topological superconductivity of spin-3/2 carriers in a three-dimensional doped Luttinger semimetal
- Freezing and decorated Poisson point processes
- Extremal statistics for stochastic resetting systems
- Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
- Exact asymptotics of the freezing transition of a logarithmically correlated random energy model
- High values of disorder-generated multifractals and logarithmically correlated processes
- Hessian spectrum at the global minimum of high-dimensional random landscapes
- Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal noise
- An Exact Solution for Spin and Charge Correlations in Quantum Dots: The Effect of Level Fluctuations and Zeeman Splitting
- Spectral properties of the trap model on sparse networks
- A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise
- Description of turbulent dynamics in the interstellar medium: multifractal/microcanonical analysis I. Application to Herschel observations of the Musca filament
- A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
- Statistics of extremes in eigenvalue-counting staircases
- Extremal statistics for a resetting Brownian motion before its first-passage time
- Extreme values of CUE characteristic polynomials: a numerical study
- Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions
- One step replica symmetry breaking and extreme order statistics of logarithmic REMs
- Non-Gaussian effects and multifractality in the Bragg glass
- Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
- Towards rigorous analysis of the Levitov-Mirlin-Evers recursion
- Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks
- Directed random walks on hierarchic trees with continuous branching: a renormalization group approach
- Multifractal analysis of the electronic states in the Fibonacci superlattice under weak electric fields
- The calculation of multi-fractal properties of directed random walks on hierarchic trees with continuous branching
- A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures
- On the existence of a glass transition in a Random Energy Model
- Optimization landscape in the simplest constrained random least-square problem
- Disordered statistical physics in low dimensions: extremes, glass transition, and localization
- Pseudo quantum advantages in perceptron storage capacity