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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 52e65bd..a285c97 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -591,6 +591,44 @@ complexity in the ground state are d_d=\hat\beta r_d \end{align} +\subsection{Full RSB} + +This reasoning applies equally well to FRSB systems. Using standard +manipulations (Appendix B), one finds also a continuous version of the +supersymmetric complexity +\begin{equation} \label{eq:functional.action} + \Sigma(E,\mu) + =\mathcal D(\mu) + + + \hat\beta E-\mu r_d + +\frac12\hat\beta r_df'(1)+\frac12r_d^2f''(1)+\frac12\log r_d^2 + +\frac12\int_0^1dq\,\left( + \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+r_d/\hat\beta} + \right) +\end{equation} +where $\chi(q)=\int_1^qdq'\int_0^{q'}dq''\,P(q)$, as in the equilibrium case. +Though the supersymmetric solution leads to a nice tractable expression, it +turn out to be useful only at one point of interest: the ground state. Indeed, +we know from the equilibrium that in the ground state $\chi$ is continuous in +the whole range of $q$. Therefore, the saddle solution found by extremizing +\begin{equation} + 0=\frac{\delta\Sigma}{\delta\chi(q)}=\frac12\hat\beta^2f''(q)-\frac12\frac1{(\chi(q)+r_d/\hat\beta)^2} +\end{equation} +given by +\begin{equation} + \chi(q)=\frac1{\hat\beta}\left(f''(q)^{-1/2}-r_d\right) +\end{equation} +is correct. This is only correct if it satisfies the boundary condition +$\chi(1)=0$, which requires $r_d=f''(1)^{-1/2}$. This in turn implies +$\mu=\frac1{r_d}+f''(1)r_d=\sqrt{4f''(1)}=\mu_m$. Therefore, the FRSB ground state +is exactly marginal! It is straightforward to check that these conditions are +indeed a saddle of the complexity. + +This has several implications. First, other than the ground state, there are +\emph{no} energies at which minima are most numerous; saddles always dominante. +As we will see, stable minima are numerous at energies above the ground state, +but these vanish at the ground state. + \section{Examples} \subsection{1RSB complexity} @@ -675,21 +713,7 @@ E\rangle_2$. \end{figure} \subsection{Full RSB complexity} -Using standard manipulations (Appendix B), one finds also a continuous version -\begin{equation} \label{eq:functional.action} - \begin{aligned} - \Sigma(E,\mu) - =\mathcal D(\mu) - + - \hat\beta E-\mu R_d - +\frac12(2\hat\beta R_d-D_d)f'(1)+\frac12R_d^2f''(1)+\frac12\log R_d^2 - \\ - +\frac12\int_0^1dq\,\left( - \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+R_d^2/D_d} - \right) - \end{aligned} -\end{equation} -where $\chi(q)$ is defined with respect to $Q$ exactly as in the equilibrium case. + \begin{figure} \centering @@ -738,34 +762,6 @@ where $\chi(q)$ is defined with respect to $Q$ exactly as in the equilibrium cas } \label{fig:2rsb.comparison} \end{figure} -In the case where any FRSB is present, one must work with the functional form -of the complexity \eqref{eq:functional.action}, which must be extremized with -respect to $\chi$ under the conditions that $\chi$ is concave, monotonically -decreasing, and $\chi(1)=0$, $\chi'(1)=-1$. The annealed case is found by -taking $\chi(q)=1-q$, which satisfies all of these conditions. $k$-RSB is -produced by breaking $\chi$ into $k+1$ piecewise linear segments. - -Forget for the moment these tricky requirements. The function would then be -extremized by satisfying -\begin{equation} - 0=\frac{\delta\Sigma}{\delta\lambda(q)}=\frac12\hat\beta^2f''(q)-\frac12\frac1{(\lambda(q)+R_d^2/D_d)^2} -\end{equation} -which implies the solution -\begin{equation} - \lambda^*(q)=\frac1{\hat\beta}f''(q)^{-1/2}-\frac{R_d^2}{D_d} -\end{equation} -If $f''(q)^{-1/2}$ is not concave anywhere, there is little use of this -solution. However, if it is concave everywhere it may constitute a portion of -the full solution. - -We suppose that solutions are given by -\begin{equation} - \lambda(q)=\begin{cases} - \lambda^*(q) & q<q_\textrm{max} \\ - 1-q & q\geq q_\textrm{max} - \end{cases} -\end{equation} -Continuity requires that $1-q_\textrm{max}=\lambda^*(q_\textrm{max})$. Here a picture of $\chi$ vs $C$ or $X$ vs $C$ showing limits $q_{max}$, $x_{max}$ for different energies and typical vs minima. |