diff options
Diffstat (limited to 'when_annealed.tex')
| -rw-r--r-- | when_annealed.tex | 11 |
1 files changed, 6 insertions, 5 deletions
diff --git a/when_annealed.tex b/when_annealed.tex index 91455a8..b5c9746 100644 --- a/when_annealed.tex +++ b/when_annealed.tex @@ -162,11 +162,11 @@ for the action $\mathcal S$ given by +(2\hat\beta r_\mathrm d-d_\mathrm d)f'(1) -\Delta x(2\hat\beta r_1-d_1)f'(q_1) +r_\mathrm d^2f''(1)-\Delta x\,r_1^2f''(q_1) \\ - &+\log\Big( + &+\frac1x\log\Big( \big(r_\mathrm d-\Delta x\,r_1\big)^2+d_\mathrm d\big(1-\Delta x\,q_1\big)-\Delta x\,d_1\big(1-\Delta xq_1\big) \Big) -\frac{\Delta x}x\log\Big( - (r_\mathrm d-r_1)^2+d_\mathrm d\big(1-\Delta xq_1\big) + (r_\mathrm d-r_1)^2+(d_\mathrm d-d_1)(1-q_1) \Big) \bigg] \Bigg\} @@ -224,7 +224,7 @@ bifurcating solution are known at this point, one can search for it by looking for a zero eigenvalue in the way described above. In the replica symmetric solution for points describing saddles, this line is \begin{equation} \label{eq:extremal.line} - \mu=-\frac1{z_f}\left(-2Ef'f''+\sqrt{-2f''u_f\big(E^2(f''-f')-\log\frac{f''}{f'}z_f\big)}\right) + \mu_0=-\frac1{z_f}\left(2Ef'f''+\sqrt{2f''u_f\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}\right) \end{equation} Let $M$ be the matrix of double partial derivatives of $\mathcal S$ with respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point @@ -263,7 +263,7 @@ The expression inside the inner square root is proportional to \begin{aligned} G_f &= - 2(f''-f')u_fw_f + -2(f''-f')u_fw_f -2\log^2\frac{f''}{f'}f'^2f''v_f \\ &\qquad @@ -288,7 +288,8 @@ between them. Therefore, $G_f>0$ is a necessary condition to see vanish, and enclosed inside they are found in exponential number. The red region (blown up in the inset) shows where the annealed complexity gives the wrong count and a {\oldstylenums1}\textsc{rsb} complexity in necessary. - The red points show where $\det M=0$. The gray shaded region highlights the + The red points show where $\det M=0$. The left point, which is only an + upper bound on the transition, coincides with it in this case. The gray shaded region highlights the minima, which are stationary points with $\mu>\mu_\mathrm m$. } \label{fig:complexity_35} \end{figure} |