On the phase diagram of the multiscale mean-field spin-glass

TL;DR

This work analyzes a multiscale Sherrington-Kirkpatrick spin-glass where couplings thermalize at distinct time scales, and develops a Parisi-type variational framework to describe the thermodynamic limit. The authors derive the asymptotic second moments of the scale-resolved overlaps, establish a rigorous annealed regime, prove that strong-coupling forces at least as many replica symmetry breaking levels as there are time scales, and give a sufficient condition for gaps in the order-parameter distribution. A central theme is synchronization across scales, linking each scale’s overlap to a synchronized Parisi measure and showing how multi-scale memory persists in the minimizer. The results illuminate a rich phase diagram with annealed, partial-annealing, and full-RSB regions, and provide tools for analyzing gap structures in functional order parameters for multiscale disordered systems.

Abstract

In this paper we study the phase diagram of a Sherrington-Kirkpatrick (SK) model where the couplings are forced to thermalize at different time scales. Besides being a challenging generalization of the SK model, such settings may arise naturally in physics whenever part of the many degrees of freedom of a system relaxes to equilibrium considerably faster than the others. For this model we compute the asymptotic value of the second moment of the overlap distribution. Furthermore, we provide a rigorous sufficient condition for an annealed solution to hold, identifying a high temperature, or weak coupling, region. In addition, we also prove that for sufficiently strong couplings the solution must present a number of replica symmetry breaking levels at least equal to the number of time scales already present in the multiscale model. Finally, we give a sufficient condition for the existence of gaps in the support of the functional order parameters.

Figures (3)

• Figure 1: Example of construction of the sequence $(x^\mu,\xi^\mu)$ associated to a probabilty measure $\mu$ with CDF $F_{\mu}$ plotted in blue. In this example we have $k=7$, $m_0<\zeta_0,\zeta_1<m_1$, $\zeta_2=m_2$, and $m_2<\zeta_3<1$. It results in the sequence $x^\mu=\{0,y_1,y_1,y_1,y_2,y_3,y_3\}$, that corresponds to the horizontal coordinates of the colored circles in the figure. Blue circles are due to the elements already present in the couple of sequences $(m,y)$, whereas red circles come from the newly introduced elements which form $\xi$. As apparent from this figure, repetitions occur all the times we are adding a $\zeta$ to the sequence $m$ which was not already contained in the latter.
• Figure 2: Typical limiting situation in which one of the $\zeta_\ell$'s (in this case $\zeta_2$) disappears from the cumulative distribution function. In general, $\zeta_\ell$ is not in the final limiting distribution when $x_{j_{\ell}+1}\to x_{j_\ell}$ for $j_\ell$ s.t. $\xi_{j_\ell}=\zeta_\ell$. In this plot, $r=4$, $k=6$.
• Figure 3: Synthetic representation of our findings.

Theorems & Definitions (36)

• Definition 1: Pressure per particle
• Remark 1
• Remark 2
• Remark 3
• Definition 2
• Remark 4
• Definition 3
• Theorem 1
• Definition 4: Quantile
• Proposition 2