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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 180798a..3c551d7 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -43,9 +43,9 @@ of small temperature for the lowest states, as it should. Here we consider, for definiteness, the mixed $p$-spin model, itself a particular case of the `Toy Model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds} \begin{equation} - H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} + H(s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} \end{equation} -for $\overline{J^2}=p!/2N^{p-1}$. Then +for $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$. Then \begin{equation} \overline{H(s_1)H(s_2)}=Nf\left(\frac{s_1\cdot s_2}N\right) \end{equation} @@ -637,50 +637,58 @@ for different energies and typical vs minima. \section{Interpretation} +Let $\langle A\rangle$ be average over stationary points with given $\epsilon$ and $\mu$, i.e., \begin{equation} - H(s)-h^Ts+g\xi^Ts -\end{equation} -Let $\langle A\mid\epsilon,\mu\rangle$ be average over stationary points with given $\epsilon$ and $\mu$, i.e., -\begin{equation} - \langle A\mid\epsilon,\mu\rangle + \langle A\rangle + =\frac1{\mathcal N}\sum_{\sigma}A(s_\sigma) =\frac1{\mathcal N} - \int d\nu(s\mid\epsilon,\mu)\,A(s) + \int d\nu(s)\,A(s) \end{equation} with \begin{equation} - d\nu(s\mid\epsilon,\mu)=ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)| + d\nu(s)=ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)| \end{equation} +Then \begin{equation} - \frac1N\|\langle s\mid\epsilon,\mu\rangle\|^2 - =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha\mid\epsilon,\mu)\,\left(\frac1{n(n-1)}\sum_{\alpha\neq\beta}\frac{s_\alpha^Ts_\beta}N\right) - =\lim_{n\to0}\frac1{n(n-1)}\left\langle\sum_{a\neq b}^nC_{ab}\right\rangle - =\int_0^1 dx\,c(x) + \begin{aligned} + \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle} + =\lim_{n\to0}\overline{\int\prod_\alpha^nd\nu(s_\alpha)\,\frac1{n(n-1)}\sum_{a\neq b}\left(\frac{s_a^Ts_b}N\right)^p} \\ + =\lim_{n\to0}\frac1{n(n-1)}\sum_{a\neq b}^nC^p_{ab} + =\int_0^1 dx\,c^p(x) + \end{aligned} \end{equation} \begin{equation} - \frac1N\sum_i\frac{\partial\langle s_i\rangle}{\partial h_i} - =\lim_{n\to0}\int\prod_\alpha^nd\nu(s_\alpha)\,\left(\frac1n\sum_{\alpha\beta}-i\frac{\hat s_\alpha^Ts_\beta}N\right) - =\lim_{n\to0}\frac1n\left\langle\sum_{\alpha\beta}R_{\alpha\beta}\right\rangle + \begin{aligned} + \overline{\frac1{N^p}\sum_{i_1\cdots i_p}\frac{\partial\langle s_{i_1}\cdots s_{i_p}\rangle}{\partial J^{(p)}_{i_1\cdots i_p}}} + =\lim_{n\to0}\overline{\int\prod_\alpha^nd\nu(s_\alpha)\,\frac1n\sum_{ab}\left[ + \hat\beta\left(\frac{s_a^Ts_b}N\right)^p+ + p\left(-i\frac{\hat s_a^Ts_b}N\right)\left(\frac{s_a^Ts_b}N\right)^{p-1} + \right]} \\ + =\lim_{n\to0}\frac1n\sum_{ab}(\hat\beta C_{ab}^p+pR_{ab}C_{ab}^{p-1}) + =\hat\beta+pr_d-\int_0^1dx\,c^{p-1}(x)(\hat\beta c(x)+pr(x)) + \end{aligned} +\end{equation} +In particular, when the energy is unconstrained ($\hat\beta=0$), +\begin{equation} + \frac1N\sum_i\frac{\partial\langle s_i\rangle}{\partial J_i^{(1)}} =r_d-\int_0^1dx\,r(x) \end{equation} +i.e., adding a linear field causes a response in the average saddle location proportional to $r_d$. \begin{equation} \begin{aligned} - \lim_{g\to0}\overline{\frac{\partial^2\Sigma}{\partial g^2}} - =\frac1N\lim_{g\to0}\lim_{n\to0}\frac1n\overline{\int\prod_\alpha d\nu(s_\alpha)\left(\sum_\alpha i\xi^T\hat s_\alpha\right)^2} - =\lim_{n\to0}\frac1n\int\prod_\alpha d\nu(s_\alpha)\left(\sum_{ab}-\frac{\hat s_a^T\hat s_b}N\right) \\ - =-\lim_{n\to0}\frac1n\left\langle\sum_{ab}D_{ab}\right\rangle - =-d_d+\int_0^1dx\,d(x) + \frac{\partial\Sigma}{\partial a_p} + =\frac14\lim_{n\to0}\frac1n\sum_{ab}^n\left[ + \hat\beta^2C_{ab}^p+p(2\hat\beta R_{ab}-D_{ab})C_{ab}^{p-1}+p(p-1)R_{ab}^2C_{ab}^{p-2} + \right] \end{aligned} \end{equation} - - The -meaning of $R_{ab}$ is that of a response of replica $a$ to a linear field in -replica $b$: +In particular, when the energy is unconstrained ($\hat\beta=0$), \begin{equation} - R_{ab} = \frac 1 N \sum_i \overline{\frac{\delta s_i^a}{\delta h_i^b}} + \frac{\partial\Sigma}{\partial a_1}=-\frac14\lim_{n\to0}\frac1n\sum_{ab}D_{ab}=-\frac14d_d-\frac14\int_0^1dx\,d(x) \end{equation} -The $D$ may similarly be seen as the variation of the complexity with respect to a random field. +i.e., adding a linear field decreases the complexity of solutions by an amount proportional to $d_d$. \section{Ultrametricity rediscovered} |