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

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