diff options
| -rw-r--r-- | frsb_kac-rice.tex | 22 |
1 files changed, 5 insertions, 17 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 2be5afc..4b3efab 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -329,12 +329,10 @@ Taking the saddle with respect to $\mu$ and $F_d$ yields and gives \begin{align*} \Sigma - =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\frac12\left( - n - \hat\epsilon R_df'(1) - +\hat\epsilon^2\sum_{ab} + =\epsilon\hat\epsilon+\hat\epsilon R_d f'(1)+\frac12D_df'(1)+\lim_{n\to0}\frac1n\frac12\left( + \hat\epsilon^2\sum_{ab} f(Q_{ab}) - +\log\det(\hat\epsilon R_d^{-1} Q+I) + +\log\det(-D_dR_d^{-2} Q+I) \right) \end{align*} Finally, setting $0=\Sigma$ gives @@ -431,19 +429,9 @@ and +\frac12\int_0^1dq\,\left[ \hat\epsilon^2\lambda(q)f''(q) +\frac1{\lambda(q)+R_d/\hat\epsilon} - \right]\\ - -\mu R_d-\log\left(\frac1{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)+\frac12\left(1+\frac\mu{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right) + \right]-\mu R_d+\frac12R_d^2f''(1)+\log R_d\\ + +\operatorname{Re}\left\{\frac12\left(1+\frac\mu{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)-\log\left(\frac1{2f''(1)}\left(\mu\pm\sqrt{\mu^2-4f''(1)}\right)\right)\right\} \end{align*} -All that changes is now -\[ - R_d=\frac12\left( - (1-q^*)\hat\epsilon\pm\sqrt{ - (1-q^*)\left( - (1-q^*)\hat\epsilon^2+8/[f'(1)-f'(q^*) - 2\mu/\hat\epsilon] - \right) - } - \right) -\] \section{Ultrametricity rediscovered} |