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| -rw-r--r-- | frsb_kac-rice.tex | 17 |
1 files changed, 13 insertions, 4 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 741cc80..9047178 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -26,7 +26,7 @@ \maketitle \begin{abstract} We derive the general solution for the computation of stationary points of - mean-field complex landscapes. The solution incorporates Parisi's solution + mean-field complex landscapes. It incorporates Parisi's solution for the ground state, as it should. \end{abstract} @@ -403,10 +403,19 @@ where the function $\mathcal D$ is defined by \right\} \end{aligned} \end{equation} -It follows that by fixing the trace of the Hessian, we have effectively fixed -the value of the stability $\mu$ in all replicas to the value $\mu^*$, and -therefore the index of saddles in all replicas as well. +{\color{blue} +By fixing the trace of the Hessian, we have effectively fixed +the value of the stability $\mu$ in all replicas to the value $\mu^*$.\\ + +$\bullet$ For $\mu^*<\mu_m$, this amounts to fixing the index density. The argument is that +the overwhelming majority of saddles have a semicircle distribution, the fluctuations are rarer than exponential. + +$\bullet$ For the gapped case $\mu^*>\mu_m$, there is a an exponentially small probability that $r=1,2,...$ eigenvalues detach +from the semicircle in such a way that the index is in fact $N {\cal{I}}=r$. +We shall not discuss these index fluctuations in this paper, the interested +reader may find what is needed in \cite{Auffinger_2013_Complexity} +} \subsubsection{The gradient factors} The $\delta$-functions in the remaining factor are treated by writing them in |