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| -rw-r--r-- | frsb_kac-rice.tex | 17 |
1 files changed, 16 insertions, 1 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 02724ed..04fa307 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -223,7 +223,7 @@ Integrating by parts, +\frac1{\lambda(q)+R_d/\hat\epsilon} \right] \end{align*} -for $\lambda$ concave, monotonic, and $\lambda(1)=0$ +for $\lambda$ concave, monotonic, $\lambda(1)=0$, and $\lambda'(1)=-1$ \[ 0=\frac{\partial\Sigma}{\partial R_d} =\frac12\hat\epsilon f'(1)-\frac12\frac1{\hat\epsilon}\int_0^1dq\,\frac{\lambda(q)}{[\lambda(q)+R_d/\hat\epsilon]^2} @@ -239,4 +239,19 @@ for $\lambda$ concave, monotonic, and $\lambda(1)=0$ \lambda^*(q)=\frac1{\hat\epsilon}\left[f''(q)^{-1/2}-R_d\right] \] +We suppose that solutions are given by +\begin{equation} + \lambda(q)=\begin{cases} + \lambda^*(q) & q<q^* \\ + 1-q & q\geq q^* + \end{cases} +\end{equation} +where $1-q$ guarantees the boundary conditions at $q=1$, and corresponds to the 0RSB or annealed solutions (annealed Kac-Rice is recovered by substituting in $1-q$ for $\lambda$). We will need to require that $1-q^*=\lambda^*(q^*)$, i.e., continuity. + +The two saddle points equations for $R_d$ and $\hat\epsilon$ give +\[ + 0=\frac12\hat\epsilon f'(1)-\frac12\int_0^{q_*}dq\,\left[f''(q)^{1/2}-R_df''(q)\right] + -\frac12\frac1{\hat\epsilon}\int_{q^*}^1dq\,\frac{1-q}{(1-q+R_d/\hat\epsilon)^2} +\] + \end{document} |