From 0eef1bbb39d6aa9ee0251429659f89783e6bc2e2 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 19:21:35 +0200 Subject: Made endpoints in figures more clear. --- figs/complexity_35.pdf | Bin 88097 -> 80917 bytes figs/order_plot_1.pdf | Bin 10853 -> 10478 bytes figs/order_plot_2.pdf | Bin 10851 -> 10692 bytes 3 files changed, 0 insertions(+), 0 deletions(-) diff --git a/figs/complexity_35.pdf b/figs/complexity_35.pdf index 5f1f6aa..bb6e088 100644 Binary files a/figs/complexity_35.pdf and b/figs/complexity_35.pdf differ diff --git a/figs/order_plot_1.pdf b/figs/order_plot_1.pdf index fcc8195..76b0003 100644 Binary files a/figs/order_plot_1.pdf and b/figs/order_plot_1.pdf differ diff --git a/figs/order_plot_2.pdf b/figs/order_plot_2.pdf index 173ef94..9efc1e4 100644 Binary files a/figs/order_plot_2.pdf and b/figs/order_plot_2.pdf differ -- cgit v1.2.3-70-g09d2 From 74d620a5c17a32cb90a885adff12d1c75d4ab31c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 19:40:59 +0200 Subject: Small spot changes. --- when_annealed.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/when_annealed.tex b/when_annealed.tex index 0304218..cbbe4f9 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -30,7 +30,7 @@ counts are reliably found by taking the average of the logarithm (the quenched average), which is more difficult and not often done in practice. When most stationary points are uncorrelated with each other, quenched and - anneals averages are equal. Equilibrium heuristics can guarantee when most of + annealed averages are equal. Equilibrium heuristics can guarantee when most of the lowest minima will be uncorrelated. We show that these equilibrium heuristics cannot be used to draw conclusions about other minima and saddles by producing examples among Gaussian-correlated functions on the hypersphere @@ -362,7 +362,7 @@ $e$, $g$, and $h$ are given by transition, coincides with it in this case. The gray shaded region highlights the minima, which are stationary points with $\mu\geq\mu_\mathrm m$. $E_\textrm{min}$ is marked on the plot as the lowest energy at which - extensive saddles are found. + saddles of extensive index are found. } \label{fig:complexity_35} \end{figure} @@ -450,7 +450,7 @@ of saddles $E_\mathrm{min}$. Also, these models have a transition boundary that smoothly connects $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, so $E_{\oldstylenums1\textsc{rsb}}^-$ corresponds to the lower bound of \textsc{rsb} complexity. For large enough -$s$, the range passes into minima, which is excepted as these models have +$s$, the range passes into minima, which is expected as these models have nontrivial complexity of their ground states. This also seems to correspond with the decoupling of the \textsc{rsb} solutions connected to $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, with -- cgit v1.2.3-70-g09d2 From 3ddd8d61b2664d06f42c88ce9ccfc7fd0a7bcc84 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 21:54:55 +0200 Subject: Added little markers for the E±_1RSB. MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- figs/complexity_35.pdf | Bin 80917 -> 81453 bytes figs/order_plot_1.pdf | Bin 10478 -> 12296 bytes figs/order_plot_2.pdf | Bin 10692 -> 11169 bytes 3 files changed, 0 insertions(+), 0 deletions(-) diff --git a/figs/complexity_35.pdf b/figs/complexity_35.pdf index bb6e088..20a2789 100644 Binary files a/figs/complexity_35.pdf and b/figs/complexity_35.pdf differ diff --git a/figs/order_plot_1.pdf b/figs/order_plot_1.pdf index 76b0003..54657cc 100644 Binary files a/figs/order_plot_1.pdf and b/figs/order_plot_1.pdf differ diff --git a/figs/order_plot_2.pdf b/figs/order_plot_2.pdf index 9efc1e4..8a09d73 100644 Binary files a/figs/order_plot_2.pdf and b/figs/order_plot_2.pdf differ -- cgit v1.2.3-70-g09d2 From dbd8014c44ed7ca75597c82a5aacc76a537468a2 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 22:52:09 +0200 Subject: Many small changes for the reveiwers. --- when_annealed.tex | 57 ++++++++++++++++++++++++++++++++++--------------------- 1 file changed, 35 insertions(+), 22 deletions(-) diff --git a/when_annealed.tex b/when_annealed.tex index cbbe4f9..733f8bd 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -3,7 +3,6 @@ \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? \usepackage[T1]{fontenc} % vector fonts \usepackage{amsmath,amssymb,latexsym,graphicx} -\usepackage{newtxtext,newtxmath} % Times \usepackage[dvipsnames]{xcolor} \usepackage[ colorlinks=true, @@ -106,15 +105,28 @@ Specifying the covariance function $f$ uniquely specifies the model. The series coefficients of $f$ need to be nonnnegative in order for $f$ to be a well-defined covariance. The case where $f$ is a homogeneous polynomial has been extensively studied, and corresponds to the pure spherical models of glass -physics or the spiked tensor models of statistical inference \cite{Castellani_2005_Spin-glass}. Here we will -study cases where $f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for -$\lambda\in(0,1)$, called $3+s$ models. These are examples of \emph{mixed} +physics or the spiked tensor models of statistical inference \cite{Castellani_2005_Spin-glass}. Here our examples will be models with $f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for +$\lambda\in(0,1)$, called $3+s$ models.\footnote{ + Though the examples and discussion will focus on the $3+s$ models, most + formulas (including the principal result in \eqref{eq:condition}) are valid for + arbitrary covariance functions $f$ under the condition that $f'(0)=0$, i.e., + that there is no linear field in the problem. This condition is necessary to + ensure that what we call `trivial' correlations are actually \emph{zero} + correlations: in the absence of a field, trivially correlated points on the + sphere are orthogonal. This simplifies our formulas by setting the overlap + $q_0$ between trivially correlated configurations to zero, which would + otherwise be another order parameter, but reduces the scope of this study. The + trivial overlap $q_0$ is also important in situations where a deterministic + field (or spike) is present, as in \cite{Ros_2019_Complex}, but deterministic + fields are likewise not considered here. +}These are examples of \emph{mixed} spherical models, which have been studied in the physics and statistics literature and host a zoo of complex orders and phase transitions \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical, Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous, Crisanti_2011_Statistical, BenArous_2019_Geometry, Subag_2020_Following, ElAlaoui_2020_Algorithmic}. + There are several well-established results on the equilibrium of this model. First, if the function $\chi(q)=f''(q)^{-1/2}$ is convex then it is not possible for the equilibrium solution to have nontrivial correlations between states at any @@ -215,13 +227,11 @@ When the complexity is calculated using the Kac--Rice formula and a physicists' tool set, the problem is reduced to the evaluation of an integral by the saddle point method for large $N$ \cite{Kent-Dobias_2023_How}. The complexity is given by extremizing an effective action, -\begin{equation} - \begin{aligned} - &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \\ - &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \\ - &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu) - \end{aligned} -\end{equation} +\begin{align} + &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \notag \\ + &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \notag \\ + &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu) +\end{align} for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb.action}. \begin{widetext} \begin{equation} \label{eq:1rsb.action} @@ -287,7 +297,7 @@ bifurcating solution are known at this point, we can search for it by looking for a flat direction. In the annealed solution for points describing saddles ($\mu<\mu_\mathrm m$), this line is \begin{equation} \label{eq:extremal.line} - \mu_0=-\frac1{z_f}\left(2Ef'f''+\sqrt{2f''u_f\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}\right) + \mu_0=-\frac{2Ef'f''}{z_f}-\sqrt{\frac{2f''u_f}{z_f^2}\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)} \end{equation} where we have chosen the lower branch as a convention (see Fig.~\ref{fig:complexity_35}) and where we define for brevity (here and @@ -405,27 +415,30 @@ proportional to \end{equation} If $G_f>0$, then the bifurcating solutions exist, and there are some saddles whose complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution. -Therefore, $G_f>0$ is a condition to see {\oldstylenums1}\textsc{rsb} in the +Therefore, $G_f>0$ is a sufficient condition to see at least {\oldstylenums1}\textsc{rsb} in the complexity. If $G_f<0$, then there is nowhere along the extremal line where -saddles can be described by such a complexity. The range of $3+s$ models where -$G_f$ is positive is shown in Fig.~\ref{fig:phases}. +saddles can be described by such a complexity, but this does not definitively +rule out \textsc{rsb}: the model may be unstable to different \textsc{rsb} +orders, or its phase boundary may simply not have a critical point on the extremal line. We +discuss the former possibility later in the paper. The range of $3+s$ models where $G_f$ is positive is +shown in Fig.~\ref{fig:phases}. \begin{figure*} \centering \includegraphics[width=0.29\textwidth]{figs/range_plot_1.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_2.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_3.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_4.pdf} \\ \vspace{-2em} \includegraphics[width=0.29\textwidth]{figs/range_plot_log_1.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_log_2.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_log_3.pdf} - \hspace{-3.25em} + \hspace{-3.4em} \includegraphics[width=0.29\textwidth]{figs/range_plot_log_4.pdf} \caption{ @@ -493,7 +506,7 @@ Consider a specific $H$ with \begin{aligned} H(\pmb\sigma) &=\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p} \\ - &\hspace{6pc}+\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s} + &\hspace{5pc}+\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s} \end{aligned} \end{equation} where the interaction tensors $J$ are drawn from zero-mean normal distributions -- cgit v1.2.3-70-g09d2 From e15ba4fd420290214eac3a6eec539963ad41810b Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 23:26:24 +0200 Subject: More fixes for referees. --- when_annealed.tex | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) diff --git a/when_annealed.tex b/when_annealed.tex index 733f8bd..be79c08 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -199,7 +199,7 @@ logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$. The annealed complexity has been computed for these models \cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched complexity has been computed for a couple examples which have nontrivial ground -states \cite{Kent-Dobias_2023_How}. The annealed complexity bounds the +states \cite{Crisanti_2006_Spherical ,Kent-Dobias_2023_How}. The annealed complexity bounds the complexity from above. A positive complexity indicates the presence of an exponentially large number of stationary points of the indicated kind, while a negative one means it is vanishingly unlikely they will appear. The line of @@ -257,7 +257,7 @@ for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb \end{aligned} \end{equation} where $\Delta x=1-x$ and -\begin{equation} +\begin{equation} \label{eq:hess.term} \mathcal D(\mu) =\begin{cases} \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2} @@ -295,7 +295,7 @@ a(E,\mu)=0$. Going along this line in the replica symmetric solution, the $x=q_1=1$ \cite{Kent-Dobias_2023_How}. Since all the parameters in the bifurcating solution are known at this point, we can search for it by looking for a flat direction. In the annealed solution for -points describing saddles ($\mu<\mu_\mathrm m$), this line is +points describing saddles (with $\mu^2\leq\mu_\mathrm m^2$ and therefore the simpler form of \eqref{eq:hess.term}), this line is \begin{equation} \label{eq:extremal.line} \mu_0=-\frac{2Ef'f''}{z_f}-\sqrt{\frac{2f''u_f}{z_f^2}\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)} \end{equation} @@ -464,11 +464,13 @@ smoothly connects $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, so $E_{\oldstylenums1\textsc{rsb}}^-$ corresponds to the lower bound of \textsc{rsb} complexity. For large enough $s$, the range passes into minima, which is expected as these models have -nontrivial complexity of their ground states. This also seems to correspond -with the decoupling of the \textsc{rsb} solutions connected to +nontrivial complexity of their ground states. Interestingly, this appears to +happen at precisely the value of $s$ for which nontrivial ground state +configurations appear, $s=12.403\ldots$. This also seems to correspond with the +decoupling of the \textsc{rsb} solutions connected to $E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, with -the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}. In -these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower +the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}. +In these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower bound, but sometimes it is given by the termination of the phase boundary extended from $E_{\oldstylenums1\textsc{rsb}}^+$. -- cgit v1.2.3-70-g09d2 From 956ac4caf8659c1489381b048b4671d0a4376d73 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 23 Aug 2023 23:29:49 +0200 Subject: Braketed some math in the bib. --- when_annealed.bib | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/when_annealed.bib b/when_annealed.bib index 5521002..decc756 100644 --- a/when_annealed.bib +++ b/when_annealed.bib @@ -89,7 +89,7 @@ @article{Crisanti_1992_The, author = {Crisanti, A. and Sommers, H.-J.}, - title = {The spherical $p$-spin interaction spin glass model: the statics}, + title = {The spherical {$p$}-spin interaction spin glass model: the statics}, journal = {Zeitschrift für Physik B Condensed Matter}, publisher = {Springer Science and Business Media LLC}, year = {1992}, @@ -102,7 +102,7 @@ @article{Crisanti_2004_Spherical, author = {Crisanti, A. and Leuzzi, L.}, - title = {Spherical $2+p$ Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition}, + title = {Spherical {$2+p$} Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition}, journal = {Physical Review Letters}, publisher = {American Physical Society (APS)}, year = {2004}, @@ -115,7 +115,7 @@ @article{Crisanti_2006_Spherical, author = {Crisanti, A. and Leuzzi, L.}, - title = {Spherical $2+p$ spin-glass model: An analytically solvable model with a glass-to-glass transition}, + title = {Spherical {$2+p$} spin-glass model: An analytically solvable model with a glass-to-glass transition}, journal = {Physical Review B}, publisher = {American Physical Society (APS)}, year = {2006}, @@ -180,7 +180,7 @@ @article{Folena_2020_Rethinking, author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico}, - title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed $p$-Spin Model}, + title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed {$p$}-Spin Model}, journal = {Physical Review X}, publisher = {American Physical Society}, year = {2020}, @@ -194,7 +194,7 @@ @article{Folena_2021_Gradient, author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico}, - title = {Gradient descent dynamics in the mixed $p$-spin spherical model: finite-size simulations and comparison with mean-field integration}, + title = {Gradient descent dynamics in the mixed {$p$}-spin spherical model: finite-size simulations and comparison with mean-field integration}, journal = {Journal of Statistical Mechanics: Theory and Experiment}, publisher = {IOP Publishing}, year = {2021}, -- cgit v1.2.3-70-g09d2 From 25d0fc1286a6e9b6c4185fcb6d2bd54fc41e5128 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 24 Aug 2023 00:13:11 +0200 Subject: Some changes to Fig 4 caption. --- when_annealed.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/when_annealed.tex b/when_annealed.tex index be79c08..516526b 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -481,8 +481,8 @@ extended from $E_{\oldstylenums1\textsc{rsb}}^+$. \includegraphics[width=0.95\columnwidth]{figs/order_plot_2.pdf} \caption{ - Examples of $3+14$ models where the solution - $E_{\oldstylenums1\textsc{rsb}}^-$ does and doesn't define the lower limit + Examples of $3+14$ models where the critical point + $E_{\oldstylenums1\textsc{rsb}}^-$ is and is not the lower limit of energies where \textsc{rsb} saddles are found. In both plots the red dot shows $E_{\oldstylenums1\textsc{rsb}}^-$, while the solid red lines shows the transition boundary with the \textsc{rs} complexity. The dashed black -- cgit v1.2.3-70-g09d2