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Diffstat (limited to '2-point.tex')
| -rw-r--r-- | 2-point.tex | 77 |
1 files changed, 55 insertions, 22 deletions
diff --git a/2-point.tex b/2-point.tex index c5e404e..d4bff09 100644 --- a/2-point.tex +++ b/2-point.tex @@ -77,6 +77,9 @@ The gradient and Hessian at a stationary point are then \end{align} where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivative with respect to $\mathbf s$. + +\section{Complexity} + We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure \begin{equation} d\nu_H(\mathbf s,\omega) @@ -89,12 +92,9 @@ measure conditioned on the energy density $E$ and stability $\mu$, \begin{equation} d\nu_H(\mathbf s,\omega\mid E,\mu) =d\nu_H(\mathbf s,\omega)\, - \delta\big(H(\mathbf s)-NE\big)\, + \delta\big(NE-H(\mathbf s)\big)\, \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\omega)\big) \end{equation} - -\section{Complexity} - We want the typical number of stationary points with energy density $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference stationary point of energy density $E_0$ and stability $\mu_0$. @@ -169,7 +169,7 @@ resulting from the average of the Hessians, the remaining part of the integrand has the form \begin{equation} e^{ - -Nm\hat\beta_0E_0-Nn\hat\beta_1E_1 + Nm\hat\beta_0E_0+Nn\hat\beta_1E_1 -\sum_a^m\left[(\pmb\sigma_a\cdot\hat{\pmb\sigma}_a)\mu_0 -\frac12\hat\mu_0(N-\pmb\sigma_a\cdot\pmb\sigma_a) \right] @@ -184,11 +184,11 @@ where we have introduced the linear operator \begin{equation} \mathcal O(\mathbf t) =\sum_a^m\delta(\mathbf t-\pmb\sigma_a)\left( - i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}+\hat\beta_0 + i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}-\hat\beta_0 \right) + \sum_a^n\delta(\mathbf t-\mathbf s_a)\left( - i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}+\hat\beta_1 + i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}-\hat\beta_1 \right) \end{equation} We have written the $H$-dependant terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that @@ -242,10 +242,10 @@ where \begin{equation} \begin{aligned} &\mathcal S_0(\mathcal Q_{00}) - =-\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\ + =\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\ &\quad+\frac1m\bigg\{ \frac12\sum_{ab}^m\left[ - \hat\beta_1^2f(C^{00}_{ab})-(2\hat\beta_1R^{00}_{ab}+D^{00}_{ab})f'(C^{00}_{ab})+(R_{ab}^{00})^2f''(C_{ab}^{00}) + \hat\beta_1^2f(C^{00}_{ab})+(2\hat\beta_1R^{00}_{ab}-D^{00}_{ab})f'(C^{00}_{ab})+(R_{ab}^{00})^2f''(C_{ab}^{00}) \right]+\frac12\log\det\begin{bmatrix}C^{00}&R^{00}\\R^{00}&D^{00}\end{bmatrix} \bigg\} \end{aligned} @@ -254,14 +254,14 @@ is the action for the ordinary, one-point complexity, and remainder is given by \begin{equation} \begin{aligned} &\mathcal S(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01}) - =-\hat\beta_1E_1-r^{11}_\mathrm d\mu_1-\frac12\hat\mu_1(1-c^{11}_\mathrm d)+\mathcal D(\mu_1) \\ + =\hat\beta_1E_1-r^{11}_\mathrm d\mu_1-\frac12\hat\mu_1(1-c^{11}_\mathrm d)+\mathcal D(\mu_1) \\ &\quad+\frac1n\sum_b^n\left\{-\frac12\hat\mu_{12}(q-C^{01}_{1b})+\sum_a^m\left[ - \hat\beta_0\hat\beta_1f(C^{01}_{ab})-(\hat\beta_0R^{01}_{ab}+\hat\beta_1R^{10}_{ab}+D^{01}_{ab})f'(C^{01}_{ab})+R^{01}_{ab}R^{10}_{ab}f''(C^{01}_{ab}) + \hat\beta_0\hat\beta_1f(C^{01}_{ab})+(\hat\beta_0R^{01}_{ab}+\hat\beta_1R^{10}_{ab}-D^{01}_{ab})f'(C^{01}_{ab})+R^{01}_{ab}R^{10}_{ab}f''(C^{01}_{ab}) \right]\right\} \\ &\quad+\frac1n\bigg\{ \frac12\sum_{ab}^n\left[ - \hat\beta_1^2f(C^{11}_{ab})-(2\hat\beta_1R^{11}_{ab}+D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab}) + \hat\beta_1^2f(C^{11}_{ab})+(2\hat\beta_1R^{11}_{ab}-D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab}) \right]\\ &\quad+\frac12\log\det\left( \begin{bmatrix} @@ -380,7 +380,7 @@ For these models, the saddle point parameters for the reference stationary point are well known, and take the values. \begin{align} \hat\beta_0 - &=\frac{(\epsilon_0+\mu_0)f'(1)+\epsilon_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\ + &=-\frac{(\epsilon_0+\mu_0)f'(1)+\epsilon_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\ r_\mathrm d^{00} &=\frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\ d_\mathrm d^{00} @@ -390,23 +390,56 @@ point are well known, and take the values. \right)^2 \end{align} +$(r^{00}_\mathrm d)^2+d^{00}_\mathrm d=1/f'(1)$ + Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, the diagonals of the inverse block matrix from above are simple expressions: \begin{align} - \tilde c_\mathrm d^{00}=\frac1{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d} && - \tilde r_\mathrm d^{00}=\frac{r^{00}_\mathrm d}{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d} && - \tilde d_\mathrm d^{00}=\frac{d^{00}_\mathrm d}{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d} + \tilde c_\mathrm d^{00}=f'(1) && + \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) && + \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1) \end{align} \begin{equation} - \hat\beta_2E_2-r_{22}^{(0)}\mu_2\frac12\left\{ - \hat\beta_2^2\big(f(1)-f(q_{22}^{(0)})\big) - +\left( - r_{12}^2+2\hat\beta_2r_{22}-\frac{2q_{12}r_{12}(r_{22}-r_{22}^{(0)})}{1-q_{22}^{(0)}} - \right)\big(f'(1)-f'(q_{22}^{(0)})\big) - \right\} + \begin{aligned} + &\Sigma_{12}=\mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1 + +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big) + +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q) + \\ + &+\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)+ + \frac12\Bigg\{ + \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big) + +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0) + \\ + &+\left( + (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right) + \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\ + &+\frac{1-q^2}{1-q^{11}_0}+\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1) + -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)} + +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big) + \\ + &+\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right) + \Bigg\} + \end{aligned} \end{equation} +\subsection{Most common neighbors with given overlap} + +\begin{align} + \hat\beta_1=0 && + \mu_1=2r^{11}_\mathrm df''(1) +\end{align} + +\begin{equation} + \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(4f''(1)-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q) + +O\big((1-q)^2\big) +\end{equation} +\begin{equation} + E_0^*=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0 +\end{equation} +\begin{equation} + \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}(E_0-E_0^*)(1-q)+O\big((1-q)^2\big) +\end{equation} \section{Isolated eigenvalue} |