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Diffstat (limited to 'stokes.tex')
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@@ -842,7 +842,7 @@ imaginary eigenvectors that contribute to its thimble. The matrix of eigenvectors can therefore be written $U=i^kO$ for an orthogonal matrix $O$, and with all eigenvectors canonically oriented $\det O=1$. We therefore have $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action before any Stokes points, -\begin{equation} +\begin{equation} \label{eq:real.thimble.partition.function} Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} \end{equation} @@ -1714,6 +1714,18 @@ Below $\mathcal E_1$, where the rank-1 saddle complexity vanishes, the complexit The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produces it is plotted in Fig.~\ref{fig:nearest.properties}. As the energy is brought below $\mathcal E_1$, $\epsilon_2-\epsilon_1\propto -|\epsilon_1-\mathcal E_1|^2$, $\varphi-90^\circ\propto-|\epsilon_1-\mathcal E_1|^{1/2}$, and $\Delta_\textrm{min}\propto|\epsilon_1-\mathcal E_1|$. The fact that the population of nearest neighbors has a energy lower than the stationary point gives some hope for the success of continuation involving these points: since Stokes points only lead to a change in weight when they involve upward flow from a point that already has weight, neighbors that have a lower energy won't be eligible to be involved in a Stokes line that causes a change of weight until the phase of $\beta$ has rotated almost $180^\circ$. \begin{figure} + \includegraphics{figs/neighbor_closest_energy.pdf} + + \caption{ + The energy $\epsilon_2$ of the nearest neighbor stationary points in the + complex plane to a given real stationary point of energy $\epsilon_1$. The + dashed line shows $\epsilon_2=\epsilon_1$. The nearest neighbor energy + coincides with the dashed line until $\mathcal E_1$, the energy where + rank-one saddles vanish, where it peels off. + } +\end{figure} + +\begin{figure} \includegraphics{figs/neighbor_plot.pdf} \caption{ @@ -1731,13 +1743,87 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce } \label{fig:nearest.properties} \end{figure} +\section{The {\it p}-spin spherical models: numerics} + +To study Stokes lines numerically, we approximated them by parametric curves. +If $z_0$ and $z_1$ are two stationary points of the action with +$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we +take the curve +\begin{equation} + z(t) + =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_iP_i^{(1,1)}(2t-1) +\end{equation} +where the $g$s are undetermined complex vectors and the $P_i^{(1,1)}(x)$ are the +Jacobi polynomials, orthogonal on the interval $[-1,1]$ under the weight +$(1-x)(1+x)$. The Jacobi polynomials are used because they are orthogonal with +respect to integration over precisely the term they appear inside above. These +are fixed by minimizing a cost function, which has a global minimum only for +Stokes lines. Defining +\begin{equation} + \mathcal L(t) + = 1-\frac{\operatorname{Re}[\dot z(z(t))^\dagger z'(t)]}{|\dot z(z(t))||z'(t)|} +\end{equation} +where $\dot z$ is the flow at $z$ given by \eqref{eq:flow}, this cost is given by +\begin{equation} \label{eq:cost} + \mathcal C=\int_0^1 dt\,\mathcal L(t) +\end{equation} +$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ +whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics. +Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes +lines. + +We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to +the parameter vectors $g$. Stokes lines are found or not between points by +using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$, +and approximating the cost integral by a finite sum. To sample nearby +stationary points and assess their propensity for Stokes points, we do the +following. First, a saddle-finding routine based on Newton's method is run on +the \emph{real} configuration space of the $p$-spin model. Then, a +saddle-finding routine is run on the complex configuration space in the close +vicinity of the real saddle, using random initial conditions in a slowly +increasing radius of the real stationary point. When this process finds a new +distinct stationary point, it is done. This method of sampling pairs heavily +biases the statistics we report here in favor of seeing Stokes points. + +Once a pair of nearby stationary points has been found, one real and one in the +complex plane, their energies are used to compute the phase $\theta$ necessary +to give $\beta$ in order to set their imaginary energies to the same value, the +necessary conditions for a Stokes line. A straight line (ignoring even the +constraint) is thrown between them and then minimized using the cost function +\eqref{eq:cost} for some initial $m=5$. Once a minimum is found, $m$ is +iteratively increased several times, each time minimizing the cost in between, +until $m=20$. If at some point in this process the cost blows up, indicating +that the solution is running away, the pair is thrown out; this happens +infrequently. At the end, there are several ways to asses whether a given +minimized line is a Stokes line: the value of the cost, the integrated +deviation from the constraint, the integrated deviation from the phase of the +two stationary points. Among minimized lines these values fall into +doubly-peaked histograms which well-separated prospective Stokes lines into +`good' and `bad' values for the given level of approximation $m$. + \subsection{Pure {\it p}-spin: is analytic continuation possible?} +After this work, one is motivated to ask: can analytic continuation be done in +even a simple complex model like the pure $p$-spin? Numeric and analytic +evidence indicates that the project is hopeless if ungapped stationary points +take a significant weight in the partition function, since for these Stokes +lines proliferate at even small continuation and there is no hope of tracking +them. However, for gapped stationary points we have seen compelling evidence +that suggests they will not participate in Stokes points, at least until a +large phase rotation of the parameter being continued. This gives some hope for +continuation of the low-temperature thermodynamic phase of the $p$-spin, where +weight is concentrated in precisely these points. + +Recalling our expression of the single-thimble contribution to the partition +function for a real stationary point of a real action expanded to lowest order +in large $|\beta|$ \eqref{eq:real.thimble.partition.function}, we can write for +the $p$-spin after an infintesimal rotation of $\beta$ into the complex plane +(before any Stokes points have been encountered) \begin{equation} \eqalign{ Z(\beta) &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta) \\ - &=\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma} + &\simeq\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma} |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta\mathcal S(s_\sigma)} \\ &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k) @@ -1745,17 +1831,20 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta N\epsilon} } \end{equation} -Following Derrida \cite{Derrida_1991_The}, +where $\mathcal N_\mathrm{typ}(\epsilon,k)$ is the typical number of stationary +points in a sample of the real $p$-spin model in the energy range $\epsilon$ to +$\epsilon+d\epsilon$ and with index $k$. +Following Derrida \cite{Derrida_1991_The}, this is related to the +\emph{average} number of stationary points in this range at large $N$ by \begin{equation} \mathcal N_\mathrm{typ}(\epsilon,k)=\overline\mathcal N(\epsilon,k)+\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2} \end{equation} where $\eta$ is a random, sample-dependant number of order one. This gives two -contributions to the partition function, but as we shall see it is the -fluctuations in the number of stationary points at a given energy level that -governs things at large $|\beta|$, not its total. This gives two terms to the typical partition function +terms to the typical partition function \begin{equation} Z_\mathrm{typ}=Z_A+Z_B \end{equation} +where \begin{eqnarray} \fl Z_A \simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k) @@ -1769,7 +1858,7 @@ governs things at large $|\beta|$, not its total. This gives two terms to the ty |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} \\ =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)} \end{eqnarray} -for +for functions $f_A$ and $f_B$ defined by \begin{eqnarray} f_A &=-\beta\epsilon+\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta @@ -1778,23 +1867,32 @@ for &=-\beta\epsilon+\frac12\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta +i\frac\pi2P(\lambda<0\mid\epsilon) \end{eqnarray} +and $P(\lambda<0\mid\epsilon)$ is the cumulative probability distribution of the eigenvalues of the spectrum given $\epsilon$, +\begin{equation} + P(\lambda<0\mid\epsilon)=\int_{-\infty}^0 d\lambda'\,\rho(\lambda'\mid\epsilon) +\end{equation} +and produces the macroscopic index $k/N$. Each integral will be dominated by its value near the maximum of the real part of the exponential argument. Assuming that $\epsilon<\epsilon_\mathrm{th}$, this maximum occurs at \begin{equation} - 0=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2} + 0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_A=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2} \end{equation} \begin{equation} - 0=-\operatorname{Re}\beta-\epsilon^2 + 0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_B=-\operatorname{Re}\beta-\epsilon \end{equation} -As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can only be reliably evaluated with a saddle point when either the period of oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore expect changes in behavior when $\epsilon=\epsilon_0$, the ground state energy. The temperature at which this happens is +As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can +only be reliably evaluated with a saddle point when either the period of +oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore +expect changes in behavior when $\epsilon=\mathcal E_0$, the ground state energy. +The temperature at which this happens is \begin{eqnarray} - \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_0+\frac12\frac p{p-1}\sqrt{\epsilon_0^2-\epsilon_\mathrm{th}^2}\\ - \operatorname{Re}\beta_B&=-\epsilon_0 + \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\mathcal E_0+\frac12\frac p{p-1}\sqrt{\mathcal E_0^2-\mathcal E_\mathrm{th}^2}\\ + \operatorname{Re}\beta_B&=-\mathcal E_0 \end{eqnarray} which for all $p\geq2$ has $\operatorname{Re}\beta_A>\operatorname{Re}\beta_B$. Therefore, the emergence of zeros in $Z_A$ does not lead to the emergence of zeros in the partition function as a whole, because $Z_B$ still produces a -coherent result (despite the unknown constant factor $\eta(\epsilon_0)$). It is -only at $\operatorname{Re}\beta_B=-\epsilon_0$ where both terms contributing to +coherent result (despite the unknown constant factor $\eta(\mathcal E_0)$). It is +only at $\operatorname{Re}\beta_B=-\mathcal E_0$ where both terms contributing to the partition function at large $N$ involve incoherent integrals near the maximum, and only here where the density of zeros is expected to become nonzero. @@ -1831,35 +1929,7 @@ large-$|\beta|$ saddle-point used to evaluate the thimble integrals. Taking the thimbles to the next order in $\beta$ may reveal more explicitly where Stokes points become important. -\section{The {\it p}-spin spherical models: numerics} - -To study Stokes lines numerically, we approximated them by parametric curves. -If $z_0$ and $z_1$ are two stationary points of the action with -$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we -take the curve -\begin{equation} - z(t) - =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_it^i -\end{equation} -where the $g$s are undetermined complex vectors. These are fixed by minimizing -a cost function, which has a global minimum only for Stokes lines. Defining -\begin{equation} - \mathcal L(t) - = 1-\frac{\operatorname{Re}[\dot z^*(z(t))\cdot z'(t)]}{|\dot z(z(t))||z'(t)|} -\end{equation} -this cost is given by -\begin{equation} - \mathcal C=\int_0^1 dt\,\mathcal L(t) -\end{equation} -$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$ -whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics. -Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes -lines. - -We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to -the parameter vectors $g$. Stokes lines are found or not between points by -using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$, -and approximating the cost integral by a finite sum. +\section{Conclusion} \section*{References} \bibliographystyle{unsrt} |