Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential

By Yan V. Fyodorov, Jean-Philippe Bouchaud. 2008

166 citations recorded by the source corpus.

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 particular (circular) variant of the model, and show that the integer moments of the partition function in the high-temperature phase are given by the well-known Dyson Coulomb gas integrals. The CLD freezing scenario allows one to use those moments for extracting the distribution of the free energy in both high- and low-temperature phases. In particular, it yields the full distribution of the minimal value in the potential sequence. This provides an explicit new class of extreme-value statistics for strongly correlated variables, manifestly different from the standard Gumbel class.

Subfields: Condensed Matter Physics, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1fddbaaa14ec64771855c74338b1f305fad24216

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Citations in the corpus

Extreme value statistics of correlated random variables: A pedagogical review (2019, unknown): “[281] Y.”
Freezing and Decorated Poisson Point Processes (2014, background): “Discussions on the implications of freezing in different models can be found in [28, 29, 30, 31, 32].”
Gaussian multiplicative chaos and applications: A review (2013, methodology): “The reader may consult [34, 68, 69] for an account of physics motivations and results, [19, 23, 110, 142] for rigorous results in the case of discrete models and [17, 58] for precise conjectures in the context of Gaussian multiplicative chaos theory.”
Convergence of the centered maximum of log-correlated Gaussian fields (2015, methodology): “It is not hard to verify (and this is done explicitly in [15]) that one can choose W so that the matrix R is positive definite for all N ; the resulting Gaussian field φN,v with correlation matrix R is the CLERM.”
Extrema of log-correlated random variables: Principles and Examples (2016, background): “This is the approach taken in [38, 41] for example, but there are major obstacles to overcome to make this type of arguments fully rigorous.”
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, background): “Let us mention the glassy phase of log-correlated random potentials (see [6, 19, 36, 37]) or the asymptotic expansion of the maximum of log-correlated random variables; see [17, 24].”
Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, methodology): “The conjecture was derived from precise computations of the moments of a suitable partition function and of the measure of high points, using statistical mechanics techniques developed for describing the extreme value statistics of disordered systems [34,38,39].”
Maxima of a randomized Riemann zeta function, and branching random walks (2015, result): “They argue that these coincide with those previously obtained for a logarithmically correlated Gaussian field [14].”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, methodology): “Fyodorov, Hiary and Keating gave a very precise conjecture for the maximum of {log |PN(z)|, |z| = 1} by relying on the replica method, and techniques from statistical mechanics predicting extreme values in disordered systems [15, 18, 19].”