SciPost Physics

SciPost Phys. 1, 011 (2016) · published 19 December 2016

• doi: 10.21468/SciPostPhys.1.2.011
• pdf
• BiBTeX
• RIS
• Submissions/Reports
• 

Abstract

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 distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$-noise.

• Cao et al., Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions
  Phys. Rev. E 97, 022117 (2018) [Crossref]
• Fyodorov et al., Statistics of Extremes in Eigenvalue-Counting Staircases
  Phys. Rev. Lett. 124, 210602 (2020) [Crossref]
• Schimmenti et al., Darcy's law of yield stress fluids on a treelike network
  Phys. Rev. E 108, L023102 (2023) [Crossref]
• Ostrovsky, A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures
  Ann. Henri Poincaré 19, 1043 (2018) [Crossref]
• Cao et al., Liouville Field Theory and Log-Correlated Random Energy Models
  Phys. Rev. Lett. 118, 090601 (2017) [Crossref]
• Ostrovsky, A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications
  Rev. Math. Phys. 30, 1830003 (2018) [Crossref]
• Derrida et al., Finite Size Corrections to the Parisi Overlap Function in the GREM
  J Stat Phys 172, 592 (2018) [Crossref]
• Ostrovsky, A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMC
  Prob. Math. Phys. 2, 533 (2021) [Crossref]
• Cao et al., Operator product expansion in Liouville field theory and Seiberg-type transitions in log-correlated random energy models
  Phys. Rev. E 97, 042111 (2018) [Crossref]