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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex new file mode 100644 index 0000000..02724ed --- /dev/null +++ b/frsb_kac-rice.tex @@ -0,0 +1,242 @@ +\documentclass[fleqn]{article} + +\usepackage{fullpage,amsmath,amssymb,latexsym} + +\begin{document} + +\section{Equilibrium} + +\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 +\end{equation} +$\log S_\infty=1+\log2\pi$. +\begin{align*} + \beta F= + -\frac12\log S_\infty+ + \frac12\lim_{n\to0}\frac1n\left(\beta^2nf(1)+\beta^2\sum_{i=0}^kn(x_i-x_{i+1})f(q_i) + +\log\left[ + \frac{ + 1+\sum_{i=0}^k(x_i-x_{i+1})q_i + }{ + 1+\sum_{i=1}^k(x_i-x_{i+1})q_i-x_1q_0 + } + \right]\right.\\ + +\frac n{x_1}\log\left[ + 1+\sum_{i=1}^k(x_i-x_{i+1})q_i-x_1q_0 + \right]\\ + \left.+\sum_{j=1}^kn(x_{j+1}^{-1}-x_j^{-1})\log\left[ + 1+\sum_{i=j+1}^k(x_i-x_{i+1})q_i-x_{j+1}q_j + \right] +\right) +\end{align*} + +\begin{align*} + \lim_{n\to0}\frac1n + \log\left[ + \frac{ + 1+\sum_{i=0}^k(x_i-x_{i+1})q_i + }{ + 1+\sum_{i=1}^k(x_i-x_{i+1})q_i-x_1q_0 + } + \right] + &= + \lim_{n\to0}\frac1n + \log\left[ + \frac{ + 1+\sum_{i=0}^k(x_i-x_{i+1})q_i + }{ + 1+\sum_{i=0}^k(x_i-x_{i+1})q_i-nq_0 + } + \right] \\ + &=q_0\left(1+\sum_{i=0}^k(x_i-x_{i+1})q_i\right)^{-1} +\end{align*} + + +\begin{align*} + \beta F= + -\frac12\log S_\infty+ + \frac12\left(\beta^2f(1)+\beta^2\sum_{i=0}^k(x_i-x_{i+1})f(q_i) + +q_0\left(1+\sum_{i=0}^k(x_i-x_{i+1})q_i\right)^{-1}\right. \\ + +\frac1{x_1}\log\left[ + 1+\sum_{i=1}^{k}(x_i-x_{i+1})q_i-x_1q_0 + \right]\\ + \left.+\sum_{j=1}^k(x_{j+1}^{-1}-x_j^{-1})\log\left[ + 1+\sum_{i=j+1}^{k}(x_i-x_{i+1})q_i-x_{j+1}q_j + \right] +\right) +\end{align*} +$q_0=0$ +\begin{align*} + \beta F= + -\frac12\log S_\infty+ + \frac12\left(\beta^2f(1)+\beta^2\sum_{i=0}^k(x_i-x_{i+1})f(q_i) + +\frac1{x_1}\log\left[ + 1+\sum_{i=1}^{k}(x_i-x_{i+1})q_i + \right]\right.\\ + \left.+\sum_{j=1}^k(x_{j+1}^{-1}-x_j^{-1})\log\left[ + 1+\sum_{i=j+1}^{k}(x_i-x_{i+1})q_i-x_{j+1}q_j + \right] +\right) +\end{align*} +$x_i=\tilde x_ix_k$, $x_k=y/\beta$, $q_k=1-z/\beta$ +\begin{align*} + \beta F= + -\frac12\log S_\infty+ + \frac12\left(\beta^2f(1)+\beta^2(y\beta^{-1}-1)f(1-z\beta^{-1})+y\beta\sum_{i=0}^{k-1}(\tilde x_i-\tilde x_{i+1})f(q_i)\right. \\ + +\frac\beta{\tilde x_1 y}\log\left[ + y\sum_{i=1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z-yz/\beta + \right]\\ + +\sum_{j=1}^{k-1}\frac\beta y(\tilde x_{j+1}^{-1}-\tilde x_j^{-1})\log\left[ + y\sum_{i=j+1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z-yz/\beta-y\tilde x_{j+1}q_j +\right]\\ + \left.-\frac\beta{\tilde x_1 y}\log\beta-\sum_{j=1}^{k-1}\frac\beta y(\tilde x_{j+1}^{-1}-\tilde x_j^{-1})\log\beta+(1-\beta y^{-1})\log\left[ + z/\beta + \right] +\right) +\end{align*} +\begin{align*} + \lim_{\beta\to\infty}F= + \frac12\left(yf(1)+zf'(1)+y\sum_{i=0}^{k-1}(\tilde x_i-\tilde x_{i+1})f(q_i) + +\frac1{\tilde x_1 y}\log\left[ + y\sum_{i=1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z + \right]\right.\\ + \left.+\sum_{j=1}^{k-1}\frac1 y(\tilde x_{j+1}^{-1}-\tilde x_j^{-1})\log\left[ + y\sum_{i=j+1}^{k-1}(\tilde x_i-\tilde x_{i+1})q_i+y+z-y\tilde x_{j+1}q_j + \right] + -\frac1y\log z +\right) +\end{align*} +$F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. $\tilde x_0=0$ and $\tilde x_k=1$. +\begin{equation} \label{eq:ground.state.free.energy} + \lim_{\beta\to\infty}F=\lim_{n\to0}\frac1n\frac12\left(nzf'(1)+y\sum_{ab}f(\tilde Q_{ab})+\frac1y\log\det(yz^{-1}\tilde Q+I) + \right) +\end{equation} + +\section{Kac-Rice} + +\begin{align*} + \Sigma + =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\left( + \sum_a\mu(F_{aa}-R_{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})-F_{ab}^2f''(Q_{ab}) + \right]\right.\\\left. + +\frac12\log\det\begin{bmatrix}Q&-iR\\-iR&-D\end{bmatrix} + -\log\det F + \right) +\end{align*} +\[ + 0=\frac{\partial\Sigma}{\partial R_{ab}} + =-\mu\delta_{ab}+\hat\epsilon f'(Q_{ab})+R_{ab}f''(Q_{ab})+\sum_c(R^2-DQ)^{-1}_{ac}R_{cb} +\] +\[ + 0=\frac{\partial\Sigma}{\partial D_{ab}} + =\frac12 f'(Q_{ab})-\frac12\sum_c(R^2-DQ)^{-1}_{ac}Q_{cb} +\] +The second equation implies +\[ + (R^2-DQ)^{-1}=Q^{-1}f'(Q) +\] +Insert the diagonal ansatz $R=R_dI$, $D=D_dI$. Then +\[ + 0=(R_df''(1)-\mu)I+\hat\epsilon f'(Q)+R_d(R_d^2I-D_dQ)^{-1} + =(R_df''(1)-\mu)I+\hat\epsilon f'(Q)+R_dQ^{-1}f'(Q) +\] +and +\[ + Q^{-1}f'(Q)=(I+D_df'(Q))/R_d^2 +\] +Substituting the second into the first, we have +\[ + 0=(R_df''(1)-\mu)I+\hat\epsilon f'(Q)+\frac1{R_d}(I+D_df'(Q)) +\] +\[ + 0=(R_df''(1)-\mu+R_d^{-1})I+(\hat\epsilon+D_d/R_d)f'(Q) +\] +The only way for this equation to be satisfied off the diagonal for nontrivial $Q$ is for $D_d=-R_d\hat\epsilon$. We therefore have +\begin{align*} + \Sigma + =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\left( + n\mu(F_d-R_d)+\frac12n\left[ + \hat\epsilon R_df'(1)+R_d^2f''(1)-F_d^2f''(1) + \right] + +\frac12\sum_{ab} + \hat\epsilon^2f(Q_{ab}) + ]\right.\\\left. + +\frac12\log\det(\hat\epsilon R_d^{-1} Q+I) + +n\log R_d + -n\log F_d + \right) +\end{align*} +Taking the saddle with respect to $\mu$ and $F_d$ yields +\[ + F_d=R_d +\] +\[ + \mu=R_d^{-1}(1+R_d^2f''(1)) +\] +and gives +\begin{align*} + \Sigma + =-\epsilon\hat\epsilon+\lim_{n\to0}\frac1n\frac12\left( + n + \hat\epsilon R_df'(1) + +\hat\epsilon^2\sum_{ab} + f(Q_{ab}) + +\log\det(\hat\epsilon R_d^{-1} Q+I) + \right) +\end{align*} +Finally, setting $0=\Sigma$ gives +\[ + \epsilon + =\lim_{n\to0}\frac1n\frac12\left(nR_df'(1)+\hat\epsilon\sum_{ab} + f(Q_{ab}) + +\frac1{\hat\epsilon}\log\det(\hat\epsilon R_d^{-1} Q+I) + \right) +\] +which is precisely \eqref{eq:ground.state.free.energy} with $R_d=z$ and $\hat\epsilon=y$. Therefore, a $(k-1)$-RSB ansatz in Kac-Rice will predict the correct ground state energy for a model whose equilibrium state at small temperatures is $k$-RSB. + +\section{Full} + +\begin{align*} + \lim_{n\to0}\frac1n\log\det(\hat\epsilon R_d^{-1} Q+I) + =x_1^{-1}\log\left(\hat\epsilon R_d^{-1}(1-\bar q_k)+1\right)+\int_{q_0^+}^{q_{k-1}}dq\,\mu(q)\log\left[\hat\epsilon R_d^{-1}\lambda(q)+1\right] +\end{align*} +where +\[ + \mu(q)=\frac{\partial x^{-1}(q)}{\partial q} +\] +Integrating by parts, +\begin{align*} + \lim_{n\to0}\frac1n\log\det(\hat\epsilon R_d^{-1} Q+I) + &=x_1^{-1}\log\left(\hat\epsilon R_d^{-1}(1-\bar q_k)+1\right)+\left[x^{-1}(q)\log[\hat\epsilon R_d^{-1}\lambda(q)+1]\right]_{q=q_0^+}^{q=q_{k-1}}-\frac{\hat\epsilon}{R_d}\int_{q_0^+}^{q_{k-1}}dq\,\frac{\lambda'(q)}{x(q)}\frac1{\hat\epsilon R_d^{-1}\lambda(q)+1}\\ + &=\log[\hat\epsilon R_d^{-1}\lambda(q_{k-1})+1]+\frac{\hat\epsilon}{R_d}\int_{q_0^+}^{q_{k-1}}dq\,\frac1{\hat\epsilon R_d^{-1}\lambda(q)+1} +\end{align*} + +\begin{align*} + \Sigma + =-\epsilon\hat\epsilon+ + \frac12\hat\epsilon R_df'(1) + +\frac12\int_0^1dq\,\left[ + \hat\epsilon^2\lambda(q)f''(q) + +\frac1{\lambda(q)+R_d/\hat\epsilon} + \right] +\end{align*} +for $\lambda$ concave, monotonic, and $\lambda(1)=0$ +\[ + 0=\frac{\partial\Sigma}{\partial R_d} + =\frac12\hat\epsilon f'(1)-\frac12\frac1{\hat\epsilon}\int_0^1dq\,\frac{\lambda(q)}{[\lambda(q)+R_d/\hat\epsilon]^2} +\] +\[ + 0=\frac{\partial\Sigma}{\partial\hat\epsilon} + =-\epsilon+\frac12R_d f'(1)+\hat\epsilon\int_0^1dq\,\lambda(q)f''(q)+\frac12\frac{R_d}{\hat\epsilon^2}\int_0^1dq\,\frac{\lambda(q)}{[\lambda(q)+R_d/\hat\epsilon]^2} +\] +\[ + 0=\frac{\delta\Sigma}{\delta\lambda(q)}=\frac12\hat\epsilon^2f''(q)-\frac12\frac1{(\lambda(q)+R_d/\hat\epsilon)^2} +\] +\[ + \lambda^*(q)=\frac1{\hat\epsilon}\left[f''(q)^{-1/2}-R_d\right] +\] + +\end{document} |