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| -rw-r--r-- | 2-point.tex | 62 |
1 files changed, 39 insertions, 23 deletions
diff --git a/2-point.tex b/2-point.tex index 6cf2f60..9d951e9 100644 --- a/2-point.tex +++ b/2-point.tex @@ -244,6 +244,8 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r \end{equation} \begin{figure} + \centering + \includegraphics{figs/single_complexity.pdf} \caption{ Plots of the complexity (logarithm of the number of stationary points) for the mixed spherical models studied in this paper. Energies and stabilities @@ -626,15 +628,24 @@ they take when the ordinary, 1-point complexity is calculated. For a replica symmetric complexity of the reference point, this results in \begin{align} \hat\beta_0 - &=-\frac{(E_0+\mu_0)f'(1)+E_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\ + &=-\frac{\mu_0f'(1)+E_0\big(f'(1)+f''(1)\big)}{u_f}\\ r_\mathrm d^{00} - &=\frac{\mu_0f(1)+E_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\ + &=\frac{\mu_0f(1)+E_0f'(1)}{u_f} \\ d_\mathrm d^{00} &=\frac1{f'(1)} -\left( - \frac{\mu_0f(1)+E_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} + \frac{\mu_0f(1)+E_0f'(1)}{u_f} \right)^2 \end{align} +where we define for brevity (here and elsewhere) the constants +\begin{align} + u_f=f(1)\big(f'(1)+f''(1)\big)-f'(1)^2 + && + v_f=f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2 +\end{align} +Note that because the coefficients of $f$ must be nonnegative for $f$ to +be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. In +general, $f^{(n)}(1)\geq f^{(m)}(1)$ if $n>m$. In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$, @@ -764,7 +775,7 @@ $r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is \end{aligned} \end{equation} -\subsection{Most common neighbors with given overlap} +\subsection{Expansion in the near neighborhood} The most common neighbors of a reference point are given by further extremizing the two-point complexity over the energy $E_1$ and stability $\mu_1$ of the @@ -788,9 +799,9 @@ The population of stationary points that are most common at each energy have the \end{equation} between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this most common value, the energy and stability of the most common neighbors at small $\Delta q$ are \begin{align} \label{eq:expansion.E.1} - E_1=E_0+\frac12\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\ + E_1&=E_0+\frac12\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\ \label{eq:expansion.mu.1} - \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big) + \mu_1&=\mu_0-\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big) \end{align} Therefore, whether the energy and stability of nearby points increases or decreases from that of the reference point depends only on whether the energy @@ -809,7 +820,7 @@ point any nearer. For the marginal minima, it is not clear that the same should When $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion is \begin{equation} \Sigma_{12} - =\frac12\frac{f'''(1)\big(f'(1)(f''(1)+f'''(1))-f''(1)^2\big)}{f''(1)^{3/2}\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)} + =\frac12\frac{f'''(1)v_f}{f''(1)^{3/2}u_f} \left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big) \end{equation} Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore, @@ -825,24 +836,29 @@ $\mu_1=\mu_0+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$ where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1} and \begin{equation} - \delta\mu_2=\frac{f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2}{f'(1)f''(1)^{3/4}}\sqrt{ - \frac{E_0-E_\mathrm{th}}2\frac{f'(1)\big(f'''(1)-2f''(1)\big)+2f''(1)^2}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} + \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{ + \frac{E_0-E_\mathrm{th}}2\frac{f'(1)\big(f'''(1)-2f''(1)\big)+2f''(1)^2}{u_f} } \end{equation} -Similarly, one finds that the energy lies in the range $E_1=E_0+\delta E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given by the coefficient in \eqref{eq:expansion.E.1} and +Similarly, one finds that the energy lies in the range $E_1=E_0+\delta +E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given +by the coefficient in \eqref{eq:expansion.E.1} and \begin{equation} - \delta E_2 - =\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\sqrt{ - \frac{ - \big[f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2\big]\big[ - f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big) - \big] - \big[ - f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1)) - \big] - } - {3\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)} - } + \begin{aligned} + \delta E_2 + &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg( + \frac{ + v_f + }{3u_f} + \big[ + f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1)) + \big] + \\ + &\hspace{17pc}\times + \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big) + \big] + \bigg)^\frac12 + \end{aligned} \end{equation} and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation \begin{equation} @@ -998,7 +1014,7 @@ which the Hessian is evaluated. } \end{figure} -Using the same methodology as above, the disorder-dependant terms are captured in the linear operator +Using the same methodology as above, the disorder-dependent terms are captured in the linear operator \begin{equation} \mathcal O(\mathbf t)= \sum_a^m\delta(\mathbf t-\pmb\sigma_a)(i\hat{\pmb\sigma}_a\cdot\partial_\mathbf t-\hat\beta_0) |