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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index eb940fa..3e06370 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -28,8 +28,6 @@ Although the Bray-Moore computation for the SK model was the first application o some replica symmetry breaking scheme, it turned out that the problem has been open ever since. - - to this date the program has been only complete for a subset of models here we present what we believe is the general scheme @@ -86,7 +84,7 @@ Here we consider, for definiteness, the `toy' model introduced by M\'ezard and P \section{Equilibrium} -Here we review the equilibrium solution. +Here we review the equilibrium solution. \cite{Crisanti_1992_The, Crisanti_1993_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical} \begin{equation} \beta F=\frac12\lim_{n\to0}\frac1n\left(\beta^2\sum_{ab}f(Q_{ab})+\log\det Q\right)-\frac12\log S_\infty @@ -199,28 +197,109 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. \section{Kac-Rice} +\cite{Auffinger_2012_Random, BenArous_2019_Geometry} + +\begin{equation} + \mathcal N(\epsilon, \mu) + =\int ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)-\mu s)|\det(\partial\partial H(s)-\mu I)| +\end{equation} +\begin{equation} + \Sigma(\epsilon,\mu)=\frac1N\log\mathcal N(\epsilon, \mu) +\end{equation} + \subsection{The replicated problem} -\begin{eqnarray} -&=& \Pi_a \delta(Eq_a) \; \Pi_a \left| \det_a( )\right| \delta(E_a-E(s_a))\nonumber\\ -&\rightarrow& \overline{\Pi_a \delta(Eq_a)} \; \overline{ \Pi_a \left| \det_a( )\right|\delta(E_a-E(s_a))}\nonumber\\ -\end{eqnarray} +\cite{Ros_2019_Complex} +\cite{Folena_2020_Rethinking} + +\begin{equation} + \begin{aligned} + \Sigma(\epsilon, \mu) + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\mathcal N^n(\epsilon) \\ + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)| + \end{aligned} +\end{equation} -the question of independence +the question of independence \cite{Bray_2007_Statistics} + +\begin{equation} + \begin{aligned} + \overline{\Sigma(\epsilon, \mu)} + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)} + \times + \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|} + \end{aligned} +\end{equation} + +\begin{equation} + \begin{aligned} + \mathcal D(\mu) + &=\frac1N\overline{\log|\det(\partial\partial H(s_a)-\mu I)|} + =\int d\lambda\,\rho(\lambda-\mu)\log|\lambda| \\ + &=\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{aligned} +\end{equation} +for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$. + +\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] + } + \\ + &=\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 + (\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 + ( + \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) + ) + \right] + \end{aligned} +\end{equation} all saddles versus only minima The parameters: -\begin{eqnarray} -Q_{ab}&=&\nonumber\\ -R_{ab}&=&\nonumber\\ -D_{ab}&=& -\end{eqnarray} +\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{equation} + S + =\mathcal D(\mu)+\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left( + \mu\sum_a^nR_{aa} + +\frac12\sum_{ab}\left[ + \hat\epsilon^2f(Q_{ab})-2\hat\epsilon R_{ab}f'(Q_{ab}) + +D_{ab}f'(Q_{ab})+R_{ab}^2f''(Q_{ab}) + \right] + +\frac12\log\det\begin{bmatrix}Q&R\\R&D\end{bmatrix} + \right) +\end{equation} \section{Replicated action} + \begin{align*} \Sigma =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\left( |