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| -rw-r--r-- | frsb_kac-rice.tex | 111 |
1 files changed, 86 insertions, 25 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index a8f7188..3c6276d 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -88,8 +88,77 @@ To constrain the model to the sphere, we use a Lagrange multiplier $\mu$, with t H(s)+\frac\mu2(N-s\cdot s) \end{equation} +At any critical point, the hessian is +\begin{equation} + \operatorname{Hess}H=\partial\partial H-\mu I +\end{equation} +$\partial\partial H$ is a GOE matrix with variance +\begin{equation} + \overline{(\partial_i\partial_jH)^2}=\frac1Nf''(1)\delta_{ij} +\end{equation} +and therefore its spectrum is given by the Wigner semicircle with radius $\sqrt{4f''(1)}$, or +\begin{equation} + \rho(\lambda)=\frac1{\pi\sqrt{f''(1)}}\sqrt{\lambda^2-4f''(1)} +\end{equation} +and the spectrum of $\operatorname{Hess}H$ is this shifted by $\mu$, or $\rho(\lambda-\mu)$. + +The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one +can decide to find the complexity of saddles of a certain macroscopic index, or +of minima with a certain harmonic stiffness. When $\mu$ is taken to be within +the range $\pm\sqrt{4f''(1)}=\pm\mu_m$, the critical points are constrained to have +index $\frac12N(1-\mu/\mu_m)$. When $\mu>\mu_m$, the critical +points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally, +when $\mu=\mu_m$, the critical points are marginal minima. +\section{Main result} + +\begin{equation} + \begin{aligned} + \overline{\Sigma(\epsilon,\mu)} + =\mathcal D(\mu) + +\operatorname*{extremum}_{\substack{R_d,D_d,\hat\epsilon\in\mathbb R\\\chi\in\Lambda}} + \left\{ + \hat\epsilon\epsilon-\mu R_d + +\frac12(2\hat\epsilon R_d-D_d)f'(1)+\frac12R_d^2f''(1) + +\log R_d \right.\\\left. + +\frac12\int_0^1dq\,\left( + \hat\epsilon^2f''(q)\chi(q)+\frac1{\chi(q)+R_d^2/D_d} + \right) + \right\} + \end{aligned} +\end{equation} +where +\begin{equation} + \mathcal D(\mu) + =\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{equation} +and $\Lambda$ is the space of functions $\chi:[0,1]\to[0,1]$ which are +monotonically decreasing, convex, and have $\chi(1)=0$ and $\chi'(1)=-1$. +If there is more than one extremum of this function, choose the one with the +smallest value of $\Sigma$. The sign of the root inside $\mathcal D(\mu)$ is +negative for $\mu>0$ and positive for $\mu<0$. + +The $k$-RSB ansatz is equivalent to piecewise linear $\chi$ with $k+1$ +pieces, with replica symmetric or 0-RSB giving $\chi(q)=1-q$. Our other major +result is that, if the equilibrium state in the vicinity of zero temperature is +given by a $k$-RSB ansatz, then the complexity is given by a $(k-1)$-RSB +ansatz. Moreover, there is an exact correspondence between the parameters of +the equilibrium saddle point in the limit of zero temperature and those of the +complexity saddle saddle at the ground state. If the equilibrium is given by $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the complexity in the ground state are +\begin{align} + \tilde x_i=\frac1{\hat\epsilon}\lim_{\beta\to\infty}\beta x_i + && + \tilde q_i=\lim_{\beta\to\infty}q_i + && + R_d=\lim_{\beta\to\infty}\beta(1-q_k) + && + D_d=R_d\hat\epsilon +\end{align} + \section{Equilibrium} Here we review the equilibrium solution. \cite{Crisanti_1992_The, Crisanti_1993_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical} @@ -209,7 +278,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. \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)| + =\int ds\,\delta(H(s)-N\epsilon)\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) @@ -224,7 +293,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. \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)| + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)| \end{aligned} \end{equation} @@ -233,7 +302,7 @@ 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)} + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)} \times \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|} \end{aligned} @@ -251,46 +320,38 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$. all saddles versus only minima -The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one -can decide to find the complexity of saddles of a certain macroscopic index, or -of minima with a certain harmonic stiffness. When $\mu$ is taken to be within -the range $\pm2\sqrt{f''(1)}=\pm\mu_m$, the critical points are constrained to have -index $\frac12N(1-\mu/\mu_m)$. When $\mu<-\mu_m$, the critical -points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally, -when $\mu=\mu_m$, the critical points are marginal minima. - \begin{equation} - \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a) + \prod_a^n\delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a) =\int \frac{\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi} - e^{\hat\epsilon(N\epsilon-H(s_a))+\hat s_a\cdot(\partial H(s_a)-\mu s_a)} + e^{\hat\epsilon(H(s_a)-N\epsilon)+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)} \end{equation} \begin{equation} \begin{aligned} \overline{ \exp\left\{ - \sum_a^n(\hat s_a\cdot\partial_a-\hat\epsilon)H(s_a) + \sum_a^n(i\hat s_a\cdot\partial_a+\hat\epsilon)H(s_a) \right\} } &=\exp\left\{ \frac12\sum_{ab}^n - (\hat s_a\cdot\partial_a-\hat\epsilon) - (\hat s_b\cdot\partial_b-\hat\epsilon) + (i\hat s_a\cdot\partial_a+\hat\epsilon) + (i\hat s_b\cdot\partial_b+\hat\epsilon) \overline{H(s_a)H(s_b)} \right\} \\ &=\exp\left\{ \frac N2\sum_{ab}^n - (\hat s_a\cdot\partial_a-\hat\epsilon) - (\hat s_b\cdot\partial_b-\hat\epsilon) + (i\hat s_a\cdot\partial_a+\hat\epsilon) + (i\hat s_b\cdot\partial_b+\hat\epsilon) f\left(\frac{s_a\cdot s_b}N\right) \right\} \\ &\hspace{-13em}\exp\left\{ \frac N2\sum_{ab}^n \left[ \hat\epsilon^2f\left(\frac{s_a\cdot s_b}N\right) - -2\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) - +\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) - +\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right) + +2i\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) + -\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right) + -\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right) \right] \right\} \end{aligned} @@ -299,7 +360,7 @@ when $\mu=\mu_m$, the critical points are marginal minima. The parameters: \begin{align} Q_{ab}=\frac1Ns_a\cdot s_b && - R_{ab}=\frac1N\hat s_a\cdot s_b && + R_{ab}=i\frac1N\hat s_a\cdot s_b && D_{ab}=\frac1N\hat s_a\cdot\hat s_b \end{align} @@ -308,10 +369,10 @@ The parameters: =\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}) + \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} + +\frac12\log\det\begin{bmatrix}Q&-iR\\-iR&D\end{bmatrix} \right) \end{equation} |