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diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index f216171..f539fbc 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -26,11 +26,14 @@ \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \begin{abstract} - We derive the general solution for counting the stationary points of - mean-field complex landscapes. It incorporates Parisi's solution - for the ground state, as it should. Using this solution, we count the - stationary points of two models: one with multi-step replica symmetry - breaking, and one with full replica symmetry breaking. + Complexity is a measure of the number of stationary points in complex + landscapes. We derive a general solution for the complexity of mean-field + complex landscapes. It incorporates Parisi's solution for the ground state, + as it should. Using this solution, we count the stationary points of two + models: one with multi-step replica symmetry breaking, and one with full + replica symmetry breaking. These examples demonstrate the consistency of the + solution and reveal that the signature of replica symmetry breaking at high + energy densities is found in high-index saddles, not minima. \end{abstract} \maketitle @@ -64,14 +67,14 @@ in the equilibrium properties of fully connected models, the complexity has only been computed in RS cases. In this paper we share the first results for the complexity with nontrivial -hierarchy. Using a general form for the solution, we detail the structure of -landscapes with a 1RSB complexity and a full RSB complexity \footnote{The - Thouless--Anderson--Palmer (TAP) complexity is the complexity of a kind of - mean-field free energy. Because of some deep thermodynamic relationships - between the TAP complexity and the equilibrium free energy, the TAP - complexity can be computed with extensions of the equilibrium method. As a - result, the TAP complexity has been previously computed for nontrivial -hierarchical structure.}. +hierarchy. Using a general form for the solution detailed in a companion +article, we describe the structure of landscapes with a 1RSB complexity and a +full RSB complexity \footnote{The Thouless--Anderson--Palmer (TAP) complexity + is the complexity of a kind of mean-field free energy. Because of some deep + thermodynamic relationships between the TAP complexity and the equilibrium + free energy, the TAP complexity can be computed with extensions of the +equilibrium method. As a result, the TAP complexity has been previously +computed for nontrivial hierarchical structure.} \cite{Kent-Dobias_2022_How}. We study the mixed $p$-spin spherical models, with Hamiltonian \begin{equation} \label{eq:hamiltonian} @@ -275,6 +278,7 @@ transitions are listed in Table~\ref{tab:energies}. $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\ $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\ $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\ + $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\ $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline \end{tabular} \caption{ @@ -287,7 +291,8 @@ transitions are listed in Table~\ref{tab:energies}. points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the traditional threshold energy, defined by the energy at which marginal - minima become most common. $E_0$ is the ground state energy. + minima become most common. $E_\mathrm m$ is the lowest energy at which + saddles or marginal minima are found. $E_0$ is the ground state energy. } \label{tab:energies} \end{table} @@ -295,13 +300,15 @@ In this model, the RS complexity gives an inconsistent answer for the complexity of the ground state, predicting that the complexity of minima vanishes at a higher energy than the complexity of saddles, with both at a lower energy than the equilibrium ground state. The 1RSB complexity resolves -these problems, predicting the same ground state as equilibrium and with a ground state stability $\mu_0=6.480\,764\ldots>\mu_m$. It predicts that the -complexity of marginal minima (and therefore all saddles) vanishes at -$E_m=-1.287\,605\,527\ldots$, which is very slightly greater than $E_0$. Saddles -become dominant over minima at a higher energy $E_\mathrm{th}=-1.287\,575\,114\ldots$. -The 1RSB complexity transitions to a RS description for dominant stationary -points at an energy $E_1=-1.273\,886\,852\ldots$. The highest energy for which -the 1RSB description exists is $E_\mathrm{max}=-0.886\,029\,051\ldots$ +these problems, predicting the same ground state as equilibrium and with a +ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that +the complexity of marginal minima (and therefore all saddles) vanishes at +$E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become +dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity +transitions to a RS description for dominant stationary points at an energy +$E_\mathrm{dom}$. The highest energy for which the 1RSB description exists is +$E_\mathrm{max}$. The numeric values for all these energies are listed in +Table~\ref{tab:energies}. For minima, the complexity does not inherit a 1RSB description until the energy is with in a close vicinity of @@ -349,16 +356,28 @@ also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_ \end{equation} In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty=-0.53125\ldots$. -Fig.~\ref{fig:frsb.phases} -shows these trajectories, along with the phase boundaries of the complexity in -this plane. Notably, the phase boundary predicted by a perturbative expansion -correctly predicts where all of the finite $k$RSB approximations terminate. +Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model. +Notably, the phase boundary predicted by a perturbative expansion +correctly predicts where the finite $k$RSB approximations terminate. Like the 1RSB model in the previous subsection, this phase boundary is oriented such that very few, low energy, minima are described by a FRSB solution, while relatively high energy saddles of high index are also. Again, this suggests that studying the mutual distribution of high-index saddle points might give insight into lower-energy symmetry breaking in more general contexts. +We have used our solution for mean-field complexity to explore how hierarchical +RSB in equilibrium corresponds to analogous hierarchical structure in the +energy landscape. In the examples we studied, a relative minority of energy +minima are distributed in a nontrivial way, corresponding to the lowest energy +densities. On the other hand, very high-index saddles begin exhibit RSB at much +higher energy densities, on the order of the energy densities associated with +RSB transitions in equilibrium. More wore is necessary to explore this +connection, as well as whether a purely \emph{geometric} explanation can be +made for the algorithmic threshold. Applying this method to the most realistic +RSB scenario for structural glasses, the so-called 1FRSB which has features of +both 1RSB and FRSB, might yield insights about signatures that should be +present in the landscape. + \paragraph{Acknowledgements} The authors would like to thank Valentina Ros for helpful discussions. |