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diff --git a/2-point.tex b/2-point.tex index 8f54a6a..717fbaf 100644 --- a/2-point.tex +++ b/2-point.tex @@ -130,7 +130,7 @@ the dynamical transition from any starting temperature landscape structure, and namely in the statistics of stationary points of the energy. In slightly less simple models, the mixed spherical models, the story changes. -There are now a range of energies with exponentially many marginal minima. It +In these models there are a range of energies with exponentially many marginal minima. It was believed that the energy level at which these marginal minima are the most common type of stationary point would play the same role as the threshold energy in the pure models (in fact we will refer to this energy level as the @@ -152,7 +152,7 @@ More specifically, one \emph{reference} point is fixed with certain properties. Then, we compute the logarithm of the number of other points constrained to lie at a fixed overlap from the reference point. Constraining the count to points of a fixed overlap from the reference point produces constrained points with atypical properties. For -instance, when the constrained overlap is made sufficiently +instance, when the required overlap is made sufficiently large, typical constrained points tend to have an isolated eigenvalue pulled out of their spectrum, and its associated eigenvector is correlated with the direction of the reference point. Without the proximity constraint, such an @@ -170,6 +170,20 @@ at the threshold energy, while at other energies typical marginal minima are far and separated by extensive barriers. Therefore, with respect to the problem of dynamics this paper merely deepens the outstanding issues. +In \S\ref{sec:model} we define the mixed spherical models and outline some of +their important properties. In the following section \S\ref{sec:results}, we go +over the main results of this work and their interpretation. In +\S\ref{sec:complexity} we outline the calculation of the two-point complexity +and its expansion in the near-neighborhood of a reference point. Details of the +calculation of the complexity are in Appendix \ref{sec:complexity-details}. In +\S\ref{sec:eigenvalue} we introduce a method for calculating the value of an +isolated eigenvalue in the spectrum at stationary points, and outline the +calculation for the mixed spherical models. Details of this calculation are in +Appendix \ref{sec:eigenvalue-details}. Finally, we draw some conclusions about +our results in \S\ref{sec:conclusion}. For the interested reader, a comparison +between the two-point complexity and the Franz--Parisi potential in the mixed +spherical models is presented in Appendix \ref{sec:franz-parisi}. + \section{The model} \label{sec:model} @@ -195,8 +209,8 @@ where the function $f$ is defined from the coefficients $a_p$ by \begin{equation} f(q)=\frac12\sum_pa_pq^p \end{equation} -The choice of $f$ has significant effect on the form of order in the model, and -this likewise influences the geometry of stationary points. +The choice of $f$ has significant effect on the form of equilibrium order in the model, and +likewise influences the geometry of stationary points \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. To enforce the spherical constraint at stationary points, we make use of a Lagrange multiplier $\omega$. This results in the extremal problem \begin{equation} @@ -261,7 +275,7 @@ where $\partial=\frac\partial{\partial\mathbf s}$ denotes the derivative with re presence of an isolated eigenvalue. \textbf{(d)} One eigenvalue leaves the bulk spectrum of a saddle point and it remains a saddle point, but now with an eigenvector correlated with the orientation of the reference vector, so - we call this a \emph{oriented saddle}. \textbf{(e)} The same happens for + we call this an \emph{oriented saddle}. \textbf{(e)} The same happens for a minimum, and we can call it an \emph{oriented minimum}. \textbf{(f)} One eigenvalue outside a positive bulk spectrum is negative, destabilizing what would otherwise have been a stable minimum, producing an \emph{oriented @@ -313,7 +327,7 @@ stationary points with energy $E_\mathrm{th}$. $E_\mathrm{gs}$, the marginal stability $\mu_\mathrm m$, and the threshold energy $E_\mathrm{th}$. The blue line shows the location of the most common type of stationary point at each energy level. The - highlighted red region shows the approximate range of minima which attract + highlighted red region shows the approximate range of minima that attract aging dynamics from a quench to zero temperature found in \cite{Folena_2020_Rethinking}. } \label{fig:complexities} @@ -325,21 +339,20 @@ In this study, our numeric examples are drawn exclusively from the model studied f_{3+4}(q)=\frac12\big(q^3+q^4\big) \end{equation} First, the ordering of its stationary points is like that of the pure spherical models, without any clustering \cite{Kent-Dobias_2023_When}. Second, properties -of its long-time dynamics have been extensively studied. Though the numeric examples all come from the $3+4$ model, the results apply to any model sharing its simple order. The annealed one-point +of its long-time dynamics have been extensively studied and are available for comparison. Though the numeric examples all come from the $3+4$ model, the results apply to any model sharing its simple order. The annealed one-point complexity of these models was calculated in \cite{BenArous_2019_Geometry}, and -for this model the annealed is expected to be correct. +for this model the annealed calculation is expected to be correct. -The one-point complexity of this model as a function of energy $E$ and -stability $\mu$ is plotted in Fig.~\ref{fig:complexities}. The same plot for a -pure $p$-spin model would consist of only a line, because $E$ and $\mu$ cannot +The one-point complexity of the $3+4$ model as a function of energy $E_0$ and +stability $\mu_0$ is plotted in Fig.~\ref{fig:complexities}. The same plot for a +pure $p$-spin model would consist of only a line, because $E_0$ and $\mu_0$ cannot be varied independently. Several important features of the complexity are highlighted: the energies of the ground state $E_\text{gs}$ and the threshold $E_\text{th}$, along with the line of marginal stability $\mu_\text m$. Along -the line of marginal stability, energies which attract aging dynamics from -different temperatures are highlighted in red. One might expect something -quantitative to mark the ends of this range, something that would differentiate +the line of marginal stability, energies that attract aging dynamics from +different temperatures are highlighted in red. One might expect some feature to mark the ends of this range, something that would differentiate marginal minima that support aging dynamics from those that do not. As -indicated in the introduction, the two-point complexity studied in this paper +indicated in the introduction, the two-point complexity we study in this paper does not produce such a result. \section{Results} @@ -350,13 +363,13 @@ as the logarithm of the number of stationary points with energy $E_1$ and stability $\mu_1$ that lie at an overlap $q$ with a typical reference stationary point whose energy is $E_0$ and stability is $\mu_0$. When the complexity is positive, there are exponentially many stationary points with the -given properties conditioned on the existence of the reference one. When it is +given properties conditioned on the existence of the reference point. When it is zero, there are only order-one such points, and when it is negative there are exponentially few (effectively, none). In the examples below, the boundary of zero complexity between exponentially many and few points is often highlighted, with parameter regions that have negative complexity having no color. Finally, as a result of the condition that the counted points lie with a given proximity to the reference point, their spectrum can be modified by the -presence of an isolated eigenvalue, which can change the stability as shown in +presence of an isolated eigenvalue, which can change their stability as shown in Fig.~\ref{fig:spectra}. \subsection{Barriers around deep states} @@ -399,7 +412,7 @@ Fig.~\ref{fig:min.neighborhood}. For stable minima, the qualitative results for the pure $p$-spin model continue to hold, with some small modifications \cite{Ros_2019_Complexity}. -The largest difference is the decoupling of nearby +The largest difference between the pure and mixed models is the decoupling of nearby stable points from nearby low-energy points: in the pure $p$-spin model, the left and right panels of Fig.~\ref{fig:min.neighborhood} would be identical up to a constant factor $-p$. Instead, for mixed models they differ substantially, @@ -407,23 +420,23 @@ as evidenced by the dot-dashed lines in both plots that in the pure models would correspond exactly with the solid lines. One significant consequence of this difference is the diminished significance of the threshold energy $E_\text{th}$: in the left panel, marginal minima of the threshold energy are -the most common among unconstrained points, but marginal minima of lower energy -are more common in the near vicinity of the example reference minimum, whose energy is lower than the threshold energy. In the pure models, all marginal minima are at the threshold energy. +the most common among unconstrained points with $q=0$, but marginal minima of lower energy +are more common in the vicinity of the example reference minimum. In the pure models, all marginal minima are at the threshold energy. The nearest neighbor points are always oriented saddles, sometimes saddles with an extensive index and sometimes index-one saddles (Fig.~\ref{fig:spectra}(d, f)). This is a result of the persistent presence of a negative isolated eigenvalue in the spectrum of the nearest neighbors, e.g., as in the shaded regions of Fig.~\ref{fig:min.neighborhood}. Like in the pure models, the minimum energy and -maximum stability of nearby points are not monotonic: there is a range of -overlap where the minimum energy of neighbors decreases with proximity. The -emergence of oriented index-one saddles along the line of lowest-energy states -at a given overlap occurs at the local minimum of this line, another similarity with the pure models -\cite{Ros_2019_Complexity}. It is not clear why this should be true or what implications it has for behavior. +maximum stability of nearby points are not monotonic in $q$: there is a range of +overlap where the minimum energy of neighbors decreases with overlap. The +transition from stable minima to index-one saddles along the line of lowest-energy states +occurs at its local minimum, another similarity with the pure models +\cite{Ros_2019_Complexity}. This point is interesting because it describes the properties of the nearest stable minima to the reference point. It is not clear why the local minimum of the boundary coincides with this point or what implications that has for behavior. \subsection{Grouping of saddle points} At stabilities lower than the marginal stability one finds saddles with an extensive index. Though, being unstable, saddles are not attractors of -dynamics, their properties influence out-of-equilibrium dynamics. For example, +dynamics, their properties do influence out-of-equilibrium dynamics. For example, high-index saddle points are stationed at the boundaries between different basins of attraction of gradient flow, and for a given basin the flow between adjacent saddle points defines a complex with implications for the landscape @@ -437,12 +450,10 @@ stability are at different rates: the energy difference between the reference and its neighbors shrinks like $\Delta q^2$, while the stability difference shrinks like $\Delta q$. This means that the near neighborhood of saddle points is dominated by the presence of other saddle points at very similar energy, but -varied index. It makes it impossible to draw conclusions about the way saddle -points are connected by gradient flow from the properties of nearest neighbors. -Descending between saddles must lower the index -- and therefore the stability --- and the energy, but if the energy and stability change with the same order -of magnitude the connected saddle points must lie at a macroscopic distance -from each other. +relatively variable index. Descending between saddles must simultaneously lower the index and the energy, but if the +energy and stability change with the same order of magnitude, the connected +saddle points must lie at a macroscopic distance from each other. This makes it impossible to use the properties of nearest neighbors to draw inferences about the way saddle +points are connected by gradient flow. \subsection{Geometry of marginal states} @@ -476,7 +487,7 @@ significant for aging dynamics. in their complexity. The nearest ones include oriented saddle points with an extensive number of downward directions, and oriented stable and marginal minima. Though most of the nearest states are found at higher - energies, they can be found at the same energy density as the reference + energies, they can also be found at the same energy density as the reference state. Fig.~\ref{fig:marginal.prop.thres} shows examples of the neighborhoods of these marginal states. \end{itemize} @@ -607,7 +618,7 @@ Rice_1944_Mathematical}. The basic idea is that stationary points of a function can be counted by integrating a Dirac $\delta$-function containing the function's gradient over its domain. Because the argument of the $\delta$-function is nonlinear in the integration variable, it must be weighted -by the determinant of the Jacobian of the argument, which happens to be the +by the determinant of its Jacobian, which happens to be the Hessian of the function. It is not common that this procedure can be analytically carried out for an explicit function. However, in the spherical models it can be carried out \emph{on average}. @@ -619,7 +630,7 @@ In order to lighten notation, we introduce the Kac--Rice measure \delta\big(\nabla H(\mathbf s,\omega)\big)\, \big|\det\operatorname{Hess}H(\mathbf s,\omega)\big| \end{equation} -containing the gradient and Hessian of the Hamiltonian, along with a $\delta$-function enforcing the spherical constraint. +containing the $\delta$-function of the gradient and determinant of the Hessian of the Hamiltonian, along with a $\delta$-function enforcing the spherical constraint. If integrated over configuration space, $\mathcal N_H=\int d\nu_H(\mathbf s,\omega)$ gives the total number of stationary points in the function. The Kac--Rice method has been used by in many studies to analyze the geometry of random functions \cite{Cavagna_1998_Stationary, Fyodorov_2007_Density, Bray_2007_Statistics, Kent-Dobias_2023_How}. More interesting is the @@ -650,7 +661,7 @@ therefore defined by \log\bigg(\int d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)\bigg)} \end{equation} Inside the logarithm, the measure \eqref{eq:measure.energy} is integrated with -the further condition that $\mathbf s$ has a certain overlap with $\pmb\sigma$. +the further condition that $\mathbf s$ has a certain overlap with the reference configuration $\pmb\sigma$. The entire expression is then integrated over $\pmb\sigma$ again by the Kac--Rice measure, then divided by a normalization. This is equivalent to summing the logarithm over all stationary points $\pmb\sigma$ with the given @@ -675,7 +686,7 @@ integration evaluated using the saddle point method. We must assume the form of order among the replicas $\mathbf s$ and $\pmb\sigma$, and we take them to be replica symmetric. Replica symmetry means that at the saddle point, all distinct pairs of replicas have the same overlap. This choice is well-motivated -for the $3+4$ model that is our immediate interest and other models like it. +for the $3+4$ and similar models. Details of the calculation can be found in Appendix~\ref{sec:complexity-details}. @@ -710,15 +721,15 @@ where the function $\mathcal D$ is defined in \eqref{eq:hessian.func} of Appendi It is possible to further extremize this expression over all the other variables but $q_0^{11}$, for which the saddle point conditions have a unique solution. However, the resulting expression is quite complicated and provides -no insight. In fact, the numeric root-finding problem is more stable preserving these parameters, rather than analytically eliminating them. +no insight. In fact, the numeric root-finding problem is more stable when preserving these parameters, rather than analytically eliminating them. In practice, the complexity can be calculated in two ways. First, the extremal problem can be done numerically, initializing from $q=0$ where the problem reduces to that of the single-point complexity of points with energy $E_1$ and stability $\mu_1$, which has an analytical solution. Then small steps in $q$ or other -parameters are taken to trace out the solution. This is how the data in all the plots of +parameters are taken to analytically continue the solution. This is how the data in all the plots of this paper was produced. Second, the complexity can be calculated in the near -neighborhood of a reference point by expanding in powers of small $1-q$. This expansion indicates when nearby points can be found at arbitrarily small distance, and in that case gives the form of the pseudogap in their complexity. +neighborhood of a reference point by expanding in powers of small $\Delta q=1-q$. This expansion indicates when nearby points can be found at arbitrarily small distance, and in that case gives the form of the pseudogap in their complexity. If there is no overlap gap between the reference point and its nearest neighbors, their complexity can be calculated by an expansion in $1-q$. First, @@ -730,7 +741,7 @@ conditions \hat\beta_1=0 && \mu_1=2r^{11}_\mathrm df''(1) \end{align} -where the second is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the +where the second condition is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the nearby points are saddle points or marginal minima. When these conditions are inserted into the complexity, an expansion is made in small $1-q$, and the saddle point in the remaining parameters is taken, the result is @@ -759,97 +770,38 @@ $\Delta q$ are where $v_f$ and $u_f$ are positive functionals of $f$ defined in \eqref{eq:v.and.u} of Appendix~\ref{sec:complexity-details}. The most common neighboring saddles to a reference saddle are much nearer to the reference in energy ($\Delta q^2$) than in stability ($\Delta q$). In fact, -this scaling also holds for all neighbors to a reference +this scaling also holds for all neighbors of a reference saddle, not just the most common. Because both expressions are proportional to $E_0-E_\mathrm{dom}(\mu_0)$, whether the energy and stability of nearby points increases or decreases from that of the reference point depends only on whether the energy of the reference point is above or below that of the most common population at the same -stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm m)=E_\mathrm{th}$, +stability, i.e., to the right or left of the blue line in Fig.~\ref{fig:complexities}. In particular, since $E_\mathrm{dom}(\mu_\mathrm m)=E_\mathrm{th}$, the threshold energy is also the pivot around which the points asymptotically nearby marginal minima change their properties. To examine better the population of marginal points, it is necessary to look at the next term in the series of the complexity with $\Delta q$, since the linear -coefficient becomes zero at the marginal line. This tells us something -intuitive: stable minima have an effective repulsion between points, and one -always finds a sufficiently small $\Delta q$ such that no stationary points are found any nearer. For the marginal minima, it is not clear that the same should be true. - -For marginal points with $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion for the dominant population of near neighbors is +coefficient becomes zero at the marginal line. When $\mu=\mu_\text m$, the quadratic term in the expansion for the dominant population of near neighbors is \begin{equation} \Sigma_{12} =\frac12\frac{f'''(1)v_f}{f''(1)^{3/2}u_f} \left(\sqrt{\frac{2\big[f'(1)(f'''(1)-f''(1))+f''(1)^2\big]}{f'(1)f'''(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big) \end{equation} -Note that this expression is only valid for $\mu=\mu_\mathrm m$. This coefficient is positive when $E>E_\text{th}$ and negative when $E<E_\text{th}$. Therefore, -marginal minima whose energy $E_0$ is greater than the threshold have neighbors at arbitrarily close distance with a quadratic pseudogap, while those whose energy is less than the threshold have an overlap gap. Exactly at the threshold the cubic term in the expansion is necessary; it is not insightful to share explicitly but is positive for the $3+4$ and similar models. - -\begin{figure} - \centering - \includegraphics{figs/expansion_energy.pdf} - \hspace{1em} - \includegraphics{figs/expansion_stability.pdf} - - \caption{ - Demonstration of the convergence of the $(1-q)$-expansion for marginal - reference minima. Solid lines and shaded region show are the same as in - Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$. - The dotted lines show the expansion of most common neighbors, while the - dashed lines in both plots show the expansion for the minimum and maximum - energies and stabilities found at given $q$. - } \label{fig:expansion} -\end{figure} +This coefficient is positive when $E>E_\text{th}$ and negative when $E<E_\text{th}$. Therefore, +marginal minima whose energy $E_0$ is greater than the threshold have neighbors at arbitrarily close distance with a quadratic pseudogap, while those whose energy is less than the threshold have an overlap gap. Exactly at the threshold the cubic term in the expansion is necessary; it is not insightful to share explicitly but it is positive for the $3+4$ and similar models. The properties of the nearby states above the threshold can be further quantified. Though we know from \eqref{eq:expansion.E.1} and -\eqref{eq:expansion.mu.1} that the most common nearby points are small distance +\eqref{eq:expansion.mu.1} that the most common nearby points at small distance are extensive saddle points with higher energy than the reference point, we do not know what other kinds of stationary points might also be found in close proximity. Could these marginal minima sit at very small distance from other -marginal minima? The answer, in the end, is that the very near neighbors are -exclusively extensive saddles of higher energy. Therefore, even the marginal -minima with energies above the threshold energy have overlap gaps with one -another. - -The limit of stability in which nearby points are found to marginal minima -above the threshold are given by -$\mu_1=\mu_\mathrm m+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$ -where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1} -and -\begin{equation} - \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{ - \frac{E_0-E_\mathrm{th}}2\frac{2f''(1)\big(f''(1)-f'(1)\big)+f'(1)f'''(1)}{u_f} - } -\end{equation} -Since the limits differ from the most common points at higher order in $\Delta q$, nearby points are of the same kind as the most common population. -Similarly, one finds that the energy lies in the range $E_1=E_0+\delta -E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given -by the coefficient in \eqref{eq:expansion.E.1} and -\begin{equation} - \begin{aligned} - \delta E_2 - &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg( - \frac{ - v_f - }{3u_f} - \big[ - f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))-2f''(1)^2 - \big] - \\ - &\hspace{12pc}\times - \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)-6f''(1)^2 - \big] - \bigg)^\frac12 - \end{aligned} -\end{equation} -and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation -\begin{equation} - 0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1) -\end{equation} -These predictions from the small $1-q$ expansion are compared with numeric -saddle points for the complexity of marginal minima in -Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$. +marginal minima? The answer is that the very near neighbors are +exclusively extensive saddles of higher energy. Therefore, the marginal +minima with energies above the threshold energy also have overlap gaps with one +another. These results on the range of possible neighbors are elaborated in Appendix \ref{subsec:range}. \section{Finding the isolated eigenvalue} @@ -857,9 +809,9 @@ Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$. The two-point complexity $\Sigma_{12}$ depends on the spectrum at both stationary points through the determinant of their Hessians, but only on the bulk of the -distribution. As we saw, this bulk is unaffected by the conditions of energy +distribution. This bulk is unaffected by the conditions of energy and proximity. However, these conditions give rise to small-rank perturbations -to the Hessian, which can lead a subextensive number of eigenvalues leaving the +to the Hessian, which can cause a subextensive number of eigenvalues to leave the bulk. We study the possibility of \emph{one} stray eigenvalue. We use a technique recently developed to find the smallest eigenvalue of @@ -938,7 +890,7 @@ discussed above. We must restrict the artificial spherical model to lie in the tangent plane of the `real' spherical configuration space at the point of interest, to avoid our eigenvector pointing in a direction that violates the spherical constraint. A sketch of the setup is shown in Fig.~\ref{fig:sphere}. The free energy of the artificial model given a point $\mathbf s$ -and a specific realization of the disordered Hamiltonian is +and for a specific realization of the disordered Hamiltonian is \begin{equation} \begin{aligned} \beta F_H(\beta\mid\mathbf s,\omega) @@ -964,10 +916,10 @@ giving &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1,\omega_1) \end{aligned} \end{equation} -again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability, as in the previous section. Averaging over these conditions gives +again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability, as before. Averaging over these conditions gives \begin{equation} \begin{aligned} - F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q) + F_H(\beta\mid E_0,\mu_0,E_1,\mu_1,q) &=\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\ &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu_H(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma_1) \end{aligned} @@ -975,7 +927,7 @@ again anticipating the use of replicas. Finally, the reference configuration $\p This formidable expression is now ready to be averaged over the disordered Hamiltonians $H$. Once averaged, the minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or \begin{equation} - \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)} + \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_0,\mu_0,E_1,\mu_1,q)} \end{equation} For this calculation, there are three different sets of replicated variables. Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and @@ -991,7 +943,7 @@ Appendix~\ref{sec:eigenvalue-details}. The result for the minimum eigenvalue is \lambda_\mathrm{min} =\mu_1-\left(y+\frac1yf''(1)\right) \end{equation} -where $y$ an order parameter whose value is set by the saddle-point conditions +where $y$ is an order parameter whose value is set by the saddle-point conditions \begin{align} \label{eq:eigen.conditions.main} 0=-f''(1)+y^2(1-\mathcal X^TC\mathcal X) && @@ -1008,28 +960,27 @@ in a minimum eigenvalue \lambda_\mathrm{min}=\mu_1-\sqrt{4f''(1)}=\mu_1-\mu_\mathrm m \end{equation} that corresponds with the bottom edge of the semicircle distribution. This is -the correct solution in the absence of an isolated eigenvalue. Any nontrivial -solution, corresponding to an isolated eigenvalue, must have nonzero $\mathcal +the correct solution in the absence of an isolated eigenvalue. Any +solution corresponding to the presence of an isolated eigenvalue must have nonzero $\mathcal X$. The only way to satisfy this with the second of the saddle conditions \eqref{eq:eigen.conditions.main} is for $y$ such that one of the eigenvalues of $B-yC$ is zero. Under these circumstances, if the normalized eigenvector associated with the zero eigenvector is $\hat{\mathcal X}_0$, then $\mathcal X=\|\mathcal X_0\|\hat{\mathcal X}_0$ is a solution. The magnitude $\|\mathcal -X_0\|$ of the solution is set by the first saddle point condition, namely +X_0\|$ of this solution is set by the first saddle point condition, namely \begin{equation} \|\mathcal X_0\|^2=\frac1{\hat{\mathcal X}_0^TC\hat{\mathcal X}_0}\left(1-\frac{f''(1)}{y^2}\right) \end{equation} In practice, we find that $\hat{\mathcal X}_0^TC\hat{\mathcal X}_0$ is positive -at the saddle point. Therefore, for the solution to make sense we must have -$y^2\geq f''(1)$. In practice, there is at most \emph{one} $y$ which produces a +at the saddle point. Therefore, for the solution to exist we must have +$y^2\geq f''(1)$. In practice, there is at most one $y$ which produces a zero eigenvalue of $B-yC$ and satisfies this inequality, so the solution seems to be unique. With this solution, we simultaneously find the smallest eigenvalue and information about the orientation of its associated eigenvector: namely, its -overlap $q_\mathrm{min}$ with the tangent vector that points directly toward -the reference spin. This information is encoded the order parameter vector -$\mathcal X$, and the detail of how it is computed can be found at the end of +overlap $q_\mathrm{min}$ with the tangent vector that points directly from one stationary point to the other. This information is encoded the order parameter vector +$\mathcal X$, and the details of how it is computed can be found at the end of Appendix~\ref{sec:eigenvalue-details}. The emergence of an isolated eigenvalue and its associated eigenvector are shown in Fig.~\ref{fig:isolated.eigenvalue}, for the same reference point properties that were used in @@ -1068,7 +1019,7 @@ the threshold where the nearest marginal states transition from having an isolated eigenvalue to not having one; see for instance in the right panel of Fig.~\ref{fig:marginal.prop.above} that the grey region vanishes. One might reason that this could change the connectivity of nearby marginal-like states and -thereby the aging dynamics. However, these energies are not close to the limits +thereby the aging dynamics. However, the energies where these changes occur are not close to the limits of aging dynamics measured by \cite{Folena_2020_Rethinking}, so that reasoning is wrong. \section{Conclusion} @@ -1147,7 +1098,7 @@ INFN. \label{sec:complexity-details} The two-point complexity defined in \eqref{eq:complexity.definition} consists -of the average over integrals consisting of products of Dirac +of the average over integrals containing of products of Dirac $\delta$-functions and determinants of Hessians. To compute it, we first split the factors into two groups: one group that contains any dependence on the Hessian (the determinants and the $\delta$-functions fixing the stabilities) @@ -1211,7 +1162,7 @@ Therefore, each of the Lagrange multipliers is fixed to one of the stabilities $ \end{cases} \end{aligned} \end{equation} -and the full factor due to the Hessians can be written +and using it the full factor due to the Hessians can be written \begin{equation} e^{Nm\mathcal D(\mu_0)+Nn\mathcal D(\mu_1)}\left[\prod_a^m\delta(N\mu_0-N\varsigma_a)\right]\left[\prod_a^n\delta(N\mu_1-N\omega_a)\right] \end{equation} @@ -1224,7 +1175,7 @@ The other factors consist of $\delta$-functions of the gradient and $\delta$-fun \delta\big(\nabla H(\mathbf s,\mu_1)\big) =\int\frac{d\hat{\mathbf s}}{(2\pi)^N}e^{i\hat{\mathrm s}\cdot\nabla H(\mathbf s,\mu_1)} \end{equation} -replaces the $\delta$-function over the gradient by introducing the auxiliary field $\hat{\mathbf s}$. Carrying out such a transformation to each of the remaining factors gives an exponential integrand of the form +replaces a $\delta$-function of the gradient by introducing the auxiliary field $\hat{\mathbf s}$. Carrying out such a transformation to each of the remaining factors gives an exponential integrand of the form \begin{equation} e^{ Nm\hat\beta_0E_0+Nn\hat\beta_1E_1 @@ -1252,7 +1203,7 @@ where we have introduced the linear operator consolidating all of the $H$-dependent terms. Here the $\hat\beta$s are the fields auxiliary to the energy constraints, the $\hat\mu$s are auxiliary to the spherical and overlap constraints, and the -$\hat{\pmb\sigma}$s and $\hat{\mathbf s}$s are auxiliary to the constraint that +$\hat{\pmb\sigma}$s and $\hat{\mathbf s}$s are auxiliary to the constraints that the gradient be zero. We have written the $H$-dependent terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that \begin{equation} @@ -1289,9 +1240,9 @@ We insert into the integral the product of $\delta$-functions enforcing these definitions, integrated over the new matrix fields, which is equivalent to multiplying by one. For example, one such factor of one is given by \begin{equation} - 1=\int dC^{00}\,\prod_{ab}^m\delta(NC^{00}_{ab}-\pmb\sigma_a\cdot\pmb\sigma_b) + 1=\int dC^{00}\,\frac1{N^{m^2}}\prod_{ab}^m\delta(NC^{00}_{ab}-\pmb\sigma_a\cdot\pmb\sigma_b) \end{equation} -Once this is done, the many scalar products appearing throughout can be +Once this is done, the many scalar products appearing throughout the integrand can be replaced by the matrix fields. The only dependence of the original vector fields is from these new $\delta$-functions. These are treated schematically in following way: let $\{\mathbf a_a\}=\{\mathbf s_a,\pmb\sigma_a,\hat{\mathbf @@ -1304,6 +1255,7 @@ matrix fields. Then the $\delta$-functions described above can be promoted to an -\frac12\mathbf a^T\hat Q\mathbf a } \end{equation} +using an auxiliary matrix field $\hat Q$. The integral over the vector fields $\mathbf a$ is Gaussian and can be evaluated, giving \begin{equation} \int d\hat Q\,e^{ @@ -1317,7 +1269,7 @@ The integral over the vector fields $\mathbf a$ is Gaussian and can be evaluated \end{equation} Finally, the integral over $\hat Q$ can be evaluated using the saddle point method, giving $\hat Q=Q^{-1}$. Therefore, the term contributed to the effective -action in the matrix fields as a result of the transformation is +action as a result of the transformation is \begin{equation} \frac12\log\det Q = @@ -1343,12 +1295,12 @@ define $c_\mathrm d^{00}$, $r_\mathrm d^{00}$, $d_\mathrm d^{00}$, $c_\mathrm d^ $d_\mathrm d^{11}$ as the value of the diagonal elements of these matrices, respectively. Note that $c_\mathrm d^{00}=c_\mathrm d^{11}=1$ due to the spherical constraint. -In this paper, we will focus on models with a replica symmetric complexity, but +In this paper, we focus on models with a replica symmetric complexity, but many of the intermediate formulae are valid for arbitrary replica symmetry -breakings. At most {\oldstylenums1}\textsc{rsb} in the equilibrium is guaranteed if the function +breakings. At most {\oldstylenums1}\textsc{rsb} in equilibrium is guaranteed if the function $\chi(q)=f''(q)^{-1/2}$ is convex \cite{Crisanti_1992_The}. The complexity at the ground state must reflect the structure of equilibrium, and therefore be replica symmetric. -Recent work has found that the complexity of saddle points can produce +Recent work has found that the complexity of saddle points can have other \textsc{rsb} orders even when the ground state is replica symmetric, but the $3+4$ model has a safely replica symmetric complexity everywhere \cite{Kent-Dobias_2023_When}. Defining the `block' fields $\mathcal Q_{00}=(\hat\beta_0, \hat\mu_0, C^{00}, @@ -1542,6 +1494,75 @@ Once these expressions are inserted into the complexity, the limits of $n$ and $m$ to zero can be taken, and the parameters from $D^{01}$ and $D^{11}$ can be extremized explicitly. The result is \eqref{eq:complexity.full} from section \ref{sec:complexity} of the main text. +\subsection{The range of energies and stabilities of nearby points} +\label{subsec:range} + +The range of parameters that result in a positive complexity is found by taking +the complexity \eqref{eq:complexity.full} and further requiring that +$\Sigma_{12}=0$. The maximum and minimum stability are then found by maximizing +this constrained expression over the energy, while the maximum and minimum +energy are found by maximizing it over the stability. In the small-$\Delta q$ +expansion outlined in \S\ref{sec:complexity}, these ranges can be computed +analytically. We share the results here for the neighbors to marginal minima +with energies greater than the threshold energy, and confirm that the +analytically computed ranges match those found numerically. + +The limit of stability in which nearby points are found to marginal minima +above the threshold are given by +$\mu_1=\mu_\mathrm m+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$ +where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1} +and +\begin{equation} + \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{ + \frac{E_0-E_\mathrm{th}}2\frac{2f''(1)\big(f''(1)-f'(1)\big)+f'(1)f'''(1)}{u_f} + } +\end{equation} +Since the limits differ from the most common points at higher order in $\Delta q$, nearby points are of the same kind as the most common population. +Similarly, one finds that the energy lies in the range $E_1=E_0+\delta +E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given +by the coefficient in \eqref{eq:expansion.E.1} and +\begin{equation} + \begin{aligned} + \delta E_2 + &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg( + \frac{ + v_f + }{3u_f} + \big[ + f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))-2f''(1)^2 + \big] + \\ + &\hspace{12pc}\times + \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)-6f''(1)^2 + \big] + \bigg)^\frac12 + \end{aligned} +\end{equation} +and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation +\begin{equation} + 0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1) +\end{equation} +These predictions from the small $1-q$ expansion are compared with numeric +saddle points for the complexity of marginal minima in +Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$. + +\begin{figure} + \centering + \includegraphics{figs/expansion_energy.pdf} + \hspace{1em} + \includegraphics{figs/expansion_stability.pdf} + + \caption{ + Demonstration of the convergence of the $(1-q)$-expansion for marginal + reference minima. Solid lines and shaded region show are the same as in + Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$. + The dotted lines show the expansion of most common neighbors, while the + dashed lines in both plots show the expansion for the minimum and maximum + energies and stabilities found at given $q$. + } \label{fig:expansion} +\end{figure} + + \section{Details of calculation for the isolated eigenvalue} \label{sec:eigenvalue-details} @@ -1552,7 +1573,7 @@ exact same reasoning. We will not repeat the details of techniques that were already reported in the previous appendix. The treatment of the factors in the average over disorder proceeds as it does -for the complexity itself in \ref{subsec:other.factors}, now with the +for the complexity in \ref{subsec:other.factors}, now with the disorder-dependent terms captured in the linear operator \begin{equation} \mathcal O(\mathbf t)= @@ -1565,7 +1586,7 @@ disorder-dependent terms captured in the linear operator that is applied to $H$ by integrating over $\mathbf t\in\mathbb R^N$. The resulting expression for the integrand produces dependencies on the scalar products in \eqref{eq:fields} and on the new scalar products involving the -tangent plane vectors $\mathbf x$, +tangent plane vectors $\mathbf x$: \begin{align} A_{ab}=\frac1N\mathbf x_a\cdot\mathbf x_b && @@ -1577,7 +1598,7 @@ tangent plane vectors $\mathbf x$, && \hat X^1_{ab}=-i\frac1N\hat{\mathbf s}_a\cdot\mathbf x_b \end{align} -Replacing the original variables using a Hubbard--Stratonovich transformation then proceeds as for the complexity in subsection \ref{subsec:hubbard.strat}. +Replacing the original variables using a Hubbard--Stratonovich transformation then proceeds like it did for the complexity in subsection \ref{subsec:hubbard.strat}. Defining as before a block variable $\mathcal Q_x=(A,X^0,\hat X^0,X^1,\hat X^1)$ and consolidating the previous block variables $\mathcal Q=(\mathcal Q_{00}, \mathcal Q_{01},\mathcal Q_{11})$, we can write the minimum eigenvalue @@ -1628,13 +1649,13 @@ given by \right) \end{aligned} \end{equation} -As usual in these quenched Franz--Parisi style computations, the saddle point expressions for the variables $\mathcal Q$ in the joint limits of $m$, $n$, and $\ell$ to zero are independent of $\mathcal Q_x$, and so these quantities take the same value they do for the two-point complexity that we computed above. The saddle point conditions for the variables $\mathcal Q_x$ are then fixed by extremizing with respect to the final action. +As usual in these quenched Franz--Parisi style computations, the saddle point expressions for the variables $\mathcal Q$ in the joint limits of $m$, $n$, and $\ell$ to zero are independent of $\mathcal Q_x$, and so these quantities take the same value they do for the two-point complexity that we computed above. The saddle point conditions for the variables $\mathcal Q_x$ are found by extremizing with respect to the action once the variables $\mathcal Q$ from the complexity have been fixed. To evaluate this expression, we need a sensible ansatz for the variables $\mathcal Q_x$. The matrix $A$ we expect to be an ordinary hierarchical matrix, and since the model is a spherical 2-spin the finite but low temperature order will be replica symmetric with nonzero $a_0$. The expected form of the $X$ matrices -follows our reasoning for the 01 matrices of the previous section: namely, they +follows our reasoning for the 01 matrices of the Appendix \ref{subsec:saddle}: namely, they should have constant rows and a column structure which matches that of the level of \textsc{rsb} order associated with the degrees of freedom that parameterize the columns. Since both the reference configurations and the @@ -1693,12 +1714,12 @@ constrained configurations have replica symmetric order, we expect \hat x_1^1&\cdots&\hat x_1^1 \end{bmatrix} \end{align} -Here, the lower block of the 0 matrices is zero, because the replicas whose overlap they represent (that of the normalization of the reference configuration) have no +Here, the lower blocks of the 0 matrices are zero, because the replicas whose overlap they represent (that of the normalization of the reference configuration) have no correlation with the reference or anything else. The first row of the $X^1$ matrix needs to be zero because of the constraint that the tangent space vectors lie in the tangent plane to the sphere, and therefore have $\mathbf x_a\cdot\mathbf s_1=0$ for any $a$. This produces five parameters to deal with, which we -compile in the vector $\mathcal X=(x_0,\hat x_0,x_1\hat x_1^1,\hat x_1^0)$. +compile in the vector $\mathcal X=(x_0,\hat x_0,x_1,\hat x_1^1,\hat x_1^0)$. Inserting this ansatz is straightforward in the first part of \eqref{eq:action.eigenvalue}, but the term with $\log\det$ is again more complicated. @@ -1769,7 +1790,7 @@ where the blocks inside the inverse are given by Here, $M_{22}$ is the inverse of the matrix already analyzed as part of \eqref{eq:two-point.action}. Following our discussion of the inverses of block replica matrices above, and reasoning about their products with the rectangular -block constant matrices, things can be worked out from here using a computer +block-constant matrices, things can be worked out using a computer algebra system. For instance, the second term in $M_{11}$ contributes nothing once the appropriate limits are taken, because each contribution is proportional to $n$. |