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| -rw-r--r-- | when_annealed.tex | 16 |
1 files 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 $E_{\oldstylenums1\textsc{rsb}}^+$. |