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| -rw-r--r-- | when_annealed.tex | 10 |
1 files changed, 7 insertions, 3 deletions
diff --git a/when_annealed.tex b/when_annealed.tex index 5dc4065..df26dee 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -161,7 +161,7 @@ $s>8$ have non-convex $\chi$ and those with $s\leq8$ have convex $\chi$ independ of $\lambda$. Second, the characterization of the ground state has been made \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical, Crisanti_2011_Statistical, Auffinger_2022_The}. In the $3+s$ models we -consider, for $s>12.430...$ nontrivial ground state configurations appear in +consider, for $s>12.430...$ nontrivial ground state configurations (more than {\oldstylenums1\textsc{rsb}}) appear in a range of $\lambda$. These bounds on equilibrium order are shown in Fig.~\ref{fig:phases}, along with our result for where the complexity has nontrivial correlations between some stationary points. As evidenced in that @@ -214,7 +214,7 @@ complexity has been computed for a couple examples which have nontrivial ground 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 +negative one means it is vanishingly likely they will appear. The line of zero complexity is significant as the transition between many stationary points and none. @@ -412,7 +412,11 @@ the complexity that begins at $E_{\oldstylenums1\textsc{rsb}}^+$, the higher energy point, ends exactly at $E_{\oldstylenums1\textsc{rsb}}^-$, the lower energy point, so that these two points give the precise range of energies at which \textsc{rsb} saddles are found. An example that conforms with this -picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}. +picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}. In +that figure, the range of $\mu$ with {\oldstylenums1\textsc{rsb}} ordering at +any fixed $E$ is extremely small. With increasing $s$ the range also increases +(see the example of the $3+16$ model in \cite{Kent-Dobias_2023_How}), but we do +not have any intuition for why this is. The discriminant $\Delta_f$ inside the square root of \eqref{eq:energies} is proportional to |