Glassy phase and freezing of log-correlated Gaussian potentials

Abstract

In this paper, we consider the Gibbs measure associated to a logarithmically correlated random potential (including two dimensional free fields) at low temperature. We prove that the energy landscape freezes and enters in the so-called glassy phase. The limiting Gibbs weights are integrated atomic random measures with random intensity expressed in terms of the critical Gaussian multiplicative chaos constructed in [10,11]. This could be seen as a first rigorous step in the renormalization theory of super-critical Gaussian multiplicative chaos.

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“…(27) and (28). We now test these predictions in the zero temperature β → ∞ limit, where the gap distribution is related to the biased gap defined in (35), see eq. (37).…”

“…They are biased in the sense that events in which u min is more negative have dominating (with weight exp(−u min )) contribution to the statistics of the gap u min,1 − u min . Combining (35) and (32), we see that the first gap g 1 = V min,1 − V min satisfies…”

“…. by (44), which is a straightforward generalization of (35). SDPPPs have recently been shown to describe the minima of several models related to Branching Brownian motion [51,52], and are expected to describe minima of general logREMs.…”

“…, 2 12 , with 10 8 independent samples for each size. We measure, using (35), the cumulative distribution function of the biased gap θ (v min,1 − v min − ∆), and extrapolate the N → ∞ limit using the quadratic 1/ ln N Ansatz. We then compare this with θ (V min,1 − V min − ∆)e ∆ , measured independently in the full circular model, and extrapolated to M → ∞ using the same Ansatz in M .…”

“…Although a relation between FDC and 1RSB was hinted briefly in [31], up to now very little has been worked out explicitly in this direction. Intriguingly, recent mathematical work [35,36] provided a proof of the freezing of the distribution for F − G/β without any reference to duality, though physical approach points towards an intimate relation between the two, at least for all the models that are exactly solved (these models are sometimes referred to as integrable, although quantum/classical integrable structures behind them are not clearly identified yet). In fact, a lot of general and rigorous results concerning the extreme values of 2d GFF and the associated Boltzmann-Gibbs-Liouville measure have been obtained in recent years in mathematical literature.…”