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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index c5641a4..0da2e88 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -1395,14 +1395,39 @@ resulting magnetization of the stationary points. \section{Conclusion} -We have constructed a replica solution for the general problem of finding saddles of random mean-field landscapes, including systems -with many steps of RSB. -For systems with full RSB, we find that minima are, at all energy densities above the ground state, exponentially subdominant with respect to saddles. -The solution contains valuable geometric information that has yet to be -extracted in all detail. - -A first and very important application of the method here is to perform the calculation for high dimensional spheres, where it would give us -a clear understanding of what happens in a low-temperature realistic jamming dynamics \cite{Maimbourg_2016_Solution}. + +We have constructed a replica solution for the general problem of finding +saddles of random mean-field landscapes, including systems with many steps of +RSB. For systems with full RSB, we find that minima are, at all energy +densities above the ground state, exponentially subdominant with respect to +saddles. The solution contains valuable geometric information that has yet to +be extracted in all detail. + +A first and very important application of the method here is to perform the +calculation for high dimensional spheres, where it would give us a clear +understanding of what happens in a low-temperature realistic jamming dynamics +\cite{Maimbourg_2016_Solution}. More simply, examining the landscape of a +spherical model with a glass to glass transition from 1RSB to RS, like the +$2+4$ model when $a_4$ is larger than we have taken it in our example, might +give insight into the cases of interest for Gardner physics +\cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. In any case, our +analysis of typical 1RSB and FRSB landscapes indicates that the highest energy +signature of RSB phases at lower energies is in the overlap structure of the +high-index saddle points. Though measuring the statistics of saddle points is +difficult to imagine for experiments, this insight could find application in +simulations of glass formers, where saddle-finding methods are possible. + +A second application is to evaluate in more detail the landscape of these RSB +systems. In particular, examining the complexity of stationary points with +non-extensive indices (like rank-one saddles), the complexity of pairs of +stationary points at fixed overlap, or the complexity of barriers +\cite{Auffinger_2012_Random, Ros_2019_Complexity}. These other properties of +the landscape might shed light on the relationship between landscape RSB and +dynamical features, like the algorithmic energy $E_\mathrm{alg}$. For our 1RSB +example, because $E_\mathrm{alg}$ is just below the energy where +dominant saddles transition to a RSB complexity, we speculate that +$E_\mathrm{alg}$ may be related to the statistics of minima connected to the +saddles at this transition point. \paragraph{Acknowledgements} The authors would like to thank Valentina Ros for helpful discussions. |