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Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 70 |
1 files changed, 33 insertions, 37 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 8881f10..a8f7188 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -260,52 +260,48 @@ points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally, when $\mu=\mu_m$, the critical points are marginal minima. \begin{equation} + \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a) + =\int \frac{\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi} + e^{\hat\epsilon(N\epsilon-H(s_a))+\hat s_a\cdot(\partial H(s_a)-\mu s_a)} +\end{equation} + +\begin{equation} \begin{aligned} - \overline{\Sigma(\epsilon, \mu)} - &=\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}\overline{ - \exp\left[ - \sum_a^n - (\hat s_a\partial_a-\hat\epsilon)H(s_a) - \right] + \overline{ + \exp\left\{ + \sum_a^n(\hat s_a\cdot\partial_a-\hat\epsilon)H(s_a) + \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[ + &=\exp\left\{ \frac12\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[ + (\hat s_a\cdot\partial_a-\hat\epsilon) + (\hat s_b\cdot\partial_b-\hat\epsilon) + \overline{H(s_a)H(s_b)} + \right\} \\ + &=\exp\left\{ \frac N2\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[ + (\hat s_a\cdot\partial_a-\hat\epsilon) + (\hat s_b\cdot\partial_b-\hat\epsilon) + f\left(\frac{s_a\cdot s_b}N\right) + \right\} \\ + &\hspace{-13em}\exp\left\{ \frac N2\sum_{ab}^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] + \left[ + \hat\epsilon^2f\left(\frac{s_a\cdot s_b}N\right) + -2\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) + +\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) + +\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right) + \right] + \right\} \end{aligned} \end{equation} The parameters: -\begin{equation} - \begin{aligned} - Q_{ab}=\frac1Ns_a\cdot s_b \\ - R_{ab}=\frac1N\hat s_a\cdot s_b \\ - D_{ab}=\frac1N\hat s_a\cdot\hat s_b - \end{aligned} -\end{equation} +\begin{align} + Q_{ab}=\frac1Ns_a\cdot s_b && + R_{ab}=\frac1N\hat s_a\cdot s_b && + D_{ab}=\frac1N\hat s_a\cdot\hat s_b +\end{align} \begin{equation} S |