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Diffstat (limited to 'topology.tex')
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diff --git a/topology.tex b/topology.tex index 281831c..11f1277 100644 --- a/topology.tex +++ b/topology.tex @@ -634,7 +634,9 @@ into the structure of solutions in this regime is merited. When $M=1$ the solution manifold corresponds to the energy level set of a spherical spin glass with energy density $E=V_0/\sqrt N$. All the results from the previous sections follow, and can be translated to the spin -glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed. With a little algebra this procedure yields +glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed.\footnote{ + It is plausible that the limits of $N\to\infty$ implicit in the saddle point expansion and the limit of $\alpha\to0$ taken here do not commute, and that $M=1$ should be taken from the beginning of the calculation. However, in this case the two procedures do commute. The $\alpha\to0$ limit accomplishes only the elimination of the first term from the effective action \eqref{eq:S.m}, while following Appendix~\ref{sec:euler} with $M=1$ from the outset results in the same term not appearing in the effective action because it is of subleading order in $N$. +} With a little algebra this procedure yields \begin{align} E_\text{on}=\pm\sqrt{2f(1)} && |