diff options
Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 6013f2f..df30c38 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -119,9 +119,9 @@ when $\mu=\mu_m$, the critical points are marginal minima. =\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 + \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\log R_d^2 \right.\\\left. +\frac12\int_0^1dq\,\left( \hat\epsilon^2f''(q)\chi(q)+\frac1{\chi(q)+R_d^2/D_d} \right) @@ -283,7 +283,7 @@ $F$ 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(H(s)-N\epsilon)\delta(\partial H(s)-\mu s)|\det(\partial\partial H(s)-\mu I)| + =\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) @@ -298,7 +298,7 @@ $F$ 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(H(s_a)-N\epsilon)\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(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} @@ -307,7 +307,7 @@ 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(H(s_a)-N\epsilon)\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(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} @@ -326,37 +326,37 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$. all saddles versus only minima \begin{equation} - \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(H(s_a)-N\epsilon)+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)} + \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a) + =\int \frac{d\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi} + e^{\hat\epsilon(N\epsilon-H(s_a))+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)} \end{equation} \begin{equation} \begin{aligned} \overline{ \exp\left\{ - \sum_a^n(i\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 - (i\hat s_a\cdot\partial_a+\hat\epsilon) - (i\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 - (i\hat s_a\cdot\partial_a+\hat\epsilon) - (i\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) - +2i\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\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) + +\left(i\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right) \right] \right\} \end{aligned} @@ -365,19 +365,19 @@ all saddles versus only minima The parameters: \begin{align} Q_{ab}=\frac1Ns_a\cdot s_b && - R_{ab}=i\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} \begin{equation} S - =\mathcal D(\mu)-\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left( - -\mu\sum_a^nR_{aa} + =\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&-iR\\-iR&D\end{bmatrix} + +\frac12\log\det\begin{bmatrix}Q&iR\\iR&D\end{bmatrix} \right) \end{equation} |