Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields

摘要

We compute the distribution of the partition functions for a class ofone-dimensional Random Energy Models (REM) with logarithmically correlatedrandom potential, above and at the glass transition temperature. The randompotential sequences represent various versions of the 1/f noise generated bysampling the two-dimensional Gaussian Free Field (2dGFF) along various planarcurves. Our method extends the recent analysis of Fyodorov Bouchaud from thecircular case to an interval and is based on an analytical continuation of theSelberg integral. In particular, we unveil a {\it duality relation} satisfiedby the suitable generating function of free energy cumulants in thehigh-temperature phase. It reinforces the freezing scenario hypothesis for thatgenerating function, from which we derive the distribution of extrema for the2dGFF on the $[0,1]$ interval. We provide numerical checks of the circular andthe interval case and discuss universality and various extensions. Relevance tothe distribution of length of a segment in Liouville quantum gravity is noted.