Joint min-max distribution and Edwards-Anderson's order parameter of the circular $1 / f$-noise model

TL;DR

This paper derives exact formulas for the joint min-max distribution and Edwards-Anderson order parameter in a circular $1/f$-noise model, validating the freezing-duality conjecture and discussing implications for diffusive dynamics.

Contribution

It introduces exact calculations of key statistical quantities in the circular $1/f$-noise model using advanced mathematical techniques, confirming the freezing-duality conjecture.

Findings

Exact joint min-max distribution derived

Edwards-Anderson order parameter calculated

Validation of the freezing-duality conjecture

Abstract

We calculate the joint min--max distribution and the Edwards-Anderson's order parameter for the circular model of $1 / f$-noise. Both quantities, as well as generalisations, are obtained exactly by combining the freezing-duality conjecture and Jack-polynomial techniques. Numerical checks come with significantly improved control of finite-size effects in the glassy phase, and the results convincingly validate the freezing-duality conjecture. Application to diffusive dynamics is discussed. We also provide a formula for the pre-factor ratio of the joint/marginal Carpentier-Le Doussal tail for minimum/maximum which applies to any logarithmic random energy model.