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| -rw-r--r-- | when_annealed.tex | 57 |
1 files 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 |