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| -rw-r--r-- | frsb_kac-rice.tex | 25 |
1 files changed, 20 insertions, 5 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 099b332..ace1b28 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -241,6 +241,16 @@ the question of independence \cite{Bray_2007_Statistics} \end{equation} for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$. +all saddles versus only minima + +The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one +can decide to find the complexity of saddles of a certain macroscopic index, or +of minima with a certain harmonic stiffness. When $\mu$ is taken to be within +the range $\pm2\sqrt{f''(1)}=\pm\mu_m$, the critical points are constrained to have +index $\frac12N(1-\mu/\mu_m)$. When $\mu<-\mu_m$, the critical +points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally, +when $\mu=\mu_m$, the critical points are marginal minima. + \begin{equation} \begin{aligned} \overline{\Sigma(\epsilon, \mu)} @@ -258,23 +268,28 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$. \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} \exp\left[ \sum_{ab}^n + (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)\overline{H(s_a)H(s_b)} + \right] \\ + &=\frac1N\lim_{n\to0}\frac\partial{\partial n} + e^{nN\mathcal D(\mu)} + \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} + \exp\left[ + N\sum_{ab}^n (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)f(s_as_b/N) \right] \\ &=\frac1N\lim_{n\to0}\frac\partial{\partial n} e^{nN\mathcal D(\mu)} \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} \exp\left[ - \sum_{ab}^n + N\sum_{ab}^n ( - \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\hat s_as_bf'(s_as_b/N)+\hat s_a\hat s_bf'(s_as_b/N) - +(\hat s_as_b)^2f''(s_as_b/N) + \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\frac{\hat s_as_b}Nf'(s_as_b/N)+\frac{\hat s_a\hat s_b}Nf'(s_as_b/N) + +\left(\frac{\hat s_as_b}N\right)^2f''(s_as_b/N) ) \right] \end{aligned} \end{equation} -all saddles versus only minima - The parameters: \begin{equation} \begin{aligned} |