From cb6072491121d3482fe7dca44eeb2aa344fb5e43 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 30 Mar 2022 16:49:29 +0200 Subject: Updates to matrix ensemble section. --- stokes.tex | 178 +++++++++++++++++++++++++++++++++++++------------------------ 1 file changed, 108 insertions(+), 70 deletions(-) (limited to 'stokes.tex') diff --git a/stokes.tex b/stokes.tex index 10eda20..b98c46b 100644 --- a/stokes.tex +++ b/stokes.tex @@ -528,14 +528,14 @@ These two facts allow us to find the hessian at stationary points using a simpler technique, that of Lagrange multipliers. Suppose that our complex manifold $\tilde\Omega$ is defined by all points -$z\in\mathbb C^N$ such that $g(z)=0$ for some holomorphic function $z$. In the +$z\in\mathbb C^N$ such that $g(z)=0$ for some holomorphic function $g$. In the case of the spherical models, $g(z)=\frac12(z^Tz-N)$. Introducing the Lagrange multiplier $\mu$, we define the constrained action \begin{equation} \tilde\mathcal S(z)=\mathcal S(z)-\mu g(z) \end{equation} The condition for a stationary point is that $\partial\tilde\mathcal S=0$. This implies that, at any stationary point, -$\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\neq0$, we find the value for $\mu$ as +$\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\neq0$, we find the value for the Lagrange multiplier $\mu$ as \begin{equation} \label{eq:multiplier} \mu=\frac{\partial g^T\partial\mathcal S}{\partial g^T\partial g} \end{equation} @@ -554,7 +554,9 @@ $P\operatorname{Hess}\mathcal SP^T$. For notational simplicity we will not include this here. In order to describe the structure of thimbles, one must study the Hessian of -$\operatorname{Re}\beta\mathcal S$. We first pose the problem problem as one +$\operatorname{Re}\beta\mathcal S$, since it is the upward directions in the +flow on the real action in the vicinity of stationary points which define the thimbles in the first place. We first +pose the problem problem as one of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$, the hessian of the real part of the action with respect to these real variables is \begin{equation} \label{eq:real.hessian} @@ -608,7 +610,7 @@ the more manifestly complex way where $\operatorname{Hess}\mathcal S$ is the hessian with respect to $z$ given in \eqref{eq:complex.hessian}. -The eigenvalues and eigenvectors of the Hessian are important for evaluating +The eigenvalues and eigenvectors of the hessian are important for evaluating thimble integrals, because those associated with upward directions provide a local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb R^N$ are such that @@ -635,7 +637,7 @@ It therefore follows that the eigenvalues and vectors of the real hessian satisf \beta\operatorname{Hess}\mathcal S v=\lambda v^* \end{equation} a sort of generalized -eigenvalue problem, whose solutions are called \emph{Takagi values and vectors} \cite{Takagi_1924_On}. If we did not know the eigenvalues were real, we could +eigenvalue problem whose solutions are called the \emph{Takagi vectors} of $\operatorname{Hess}\mathcal S$ \cite{Takagi_1924_On}. If we did not know the eigenvalues were real, we could still see it from the second implied equation, $(\beta\operatorname{Hess}\mathcal S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. @@ -645,16 +647,18 @@ in its complex form is that each eigenvalue comes in a pair, since \begin{equation} \beta\operatorname{Hess}\mathcal S(iv)=i\lambda v^*=-\lambda(iv) \end{equation} -Therefore, if $\lambda$ is an eigenvalue of the hessian with eigenvector $v$, -than so is $-\lambda$, with associated eigenvector $iv$, rotated in the complex +Therefore, if $\lambda$ satisfies \eqref{eq:generalized.eigenproblem} with Takagi vector $v$, +than so does $-\lambda$, with associated Takagi $iv$, rotated in the complex plane. It follows that each stationary point has an equal number of descending -and ascending directions, e.g. the index of each stationary point is $N$. For a -stationary point in a real problem this might seem strange, because there are +and ascending directions, e.g., the index of each stationary point is $N$. For +a stationary point in a real problem this might seem strange, because there are clear differences between minima, maxima, and saddles of different index. -However, we can quickly see here that for a such a stationary point, its $N$ -real eigenvectors which determine its index in the real problem are accompanied -by $N$ purely imaginary eigenvectors, pointing into the complex plane and each -with the negative eigenvalue of its partner. +However, for a such a stationary point, its $N$ real Takagi vectors that +determine its index in the real problem are accompanied by $N$ purely imaginary +Takagi vectors, pointing into the complex plane and each with the negative +eigenvalue of its partner. A real minimum on the real manifold therefore has +$N$ downward directions alongside its $N$ upward ones, all pointing directly +into complex configuration space. The effect of changing the phase of $\beta$ is revealed by \eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and @@ -664,27 +668,27 @@ dividing both sides by $|\beta|e^{i\phi/2}$, one finds =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^* =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^* \end{equation} -Therefore, one only needs to consider solutions to the eigenproblem for the +Therefore, one only needs to consider solutions to the Takagi problem for the action alone, $\operatorname{Hess}\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the -resulting vectors by a constant phase corresponding to half the phase of +resulting Takagi vectors by a constant phase corresponding to half the phase of $\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and \ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from left to right produces a clockwise rotation in the thimbles in the complex-$\theta$ plane. -These eigenvalues and vectors can be further related to properties of the +These eigenvalues associated with the Takagi vectors can be further related to properties of the complex symmetric matrix $\beta\operatorname{Hess}\mathcal S$. Suppose that $u\in\mathbb R^N$ satisfies the eigenvalue equation \begin{equation} (\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u =\sigma u \end{equation} -for some positive real $\sigma$ (real because $(\beta\operatorname{Hess} +for some positive real $\sigma$ (because $(\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)$ is self-adjoint). The square root of these numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of $\beta\operatorname{Hess}\mathcal S$. A direct relationship between these singular -values and the eigenvalues of the hessian immediately follows by taking an -eigenvector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem}, +values and the eigenvalues of the real hessian immediately follows by taking a +Takagi vector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem}, and writing \begin{equation} \eqalign{ @@ -698,8 +702,8 @@ and writing \end{equation} Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues of the real hessian are the singular values of the complex matrix -$\beta\operatorname{Hess}\mathcal S$, and their eigenvectors coincide up to a -constant complex factor. +$\beta\operatorname{Hess}\mathcal S$, and the Takagi vectors coincide with the +eigenvectors of the singular value problem up to a constant complex factor. \subsection{The conditions for Stokes points} @@ -729,7 +733,9 @@ This is because these stationary points are not adjacent: they are separated from each other by the thimbles of other stationary points. This is a consistent story in one complex dimension, since the codimension of the thimbles is the same as the codimension of the constant imaginary energy -surface is one, and such a surface can divide space into regions. However, in higher dimensions thimbles do not have codimension high enough to divide space into regions. +surface is one, and such a surface can divide space into regions. However, in +higher dimensions thimbles do not have codimension high enough to divide space +into regions. \begin{figure} \hspace{5pc} @@ -742,14 +748,35 @@ surface is one, and such a surface can divide space into regions. However, in hi descent of the two minima, while the red and orange lines trace those of the two saddles. The location of the maxima are marked as points, but their thimbles are not shown. - } + } \label{fig:3d.thimbles} \end{figure} +Despite the fact that in higher dimensions, the level sets of constant +imaginary energy appear to usually be globally connected, thimble intersections +are still governed by a requirement for adjacency. Fig.~\ref{fig:3d.thimbles} +shows a projection of the thimbles of an $N=3$ 2-spin model, which is defined +on the sphere. Because of an inversion symmetry of the model, stationary points +on opposite sides of the sphere have identical energies, and therefore also +share the same imaginary energy. However, their thimbles (blue and green in the +figure) do not intersect. Here, they could not possibly intersect, since the +real parts of their energy are also the same, and upward flow could therefore +not connect them. + +Stokes lines, when they manifest, are persistent: if a Stokes line connects two +stationary points and the action is smoothly modified under the constraint that +the imaginary parts of the two stationary points is held equal, the Stokes line +will continue to connect them so long as the flow of a third stationary point +does not sever their connection. This implies that despite being relatively +low-dimensional surfaces of codimension $N$, thimble connections are seen with +only a codimension one tuning of parameters, modulo the topological adjacency +requirement. This means that Stokes points can generically appear when a +dimension-one curve is followed in parameter space. + \subsection{Evaluating thimble integrals} After all the work of decomposing an integral into a sub over thimbles, one eventually wants to actually evaluate it. For large $|\beta|$ and in the -absence of any Stokes points, one can come to a nice asymptotic expression. For +absence of any Stokes points, one can come to a nice asymptotic expression. For a thorough account of evaluating these integrals (including \emph{at} Stokes points), see Howls \cite{Howls_1997_Hyperasymptotics}. @@ -855,16 +882,17 @@ $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvector \section{The ensemble of symmetric complex-normal matrices} -We will now begin dealing with the implications of actions defined in very many -dimensions. We saw in \S\ref{sec:stationary.hessian} that the singular values -of the complex hessian of the action at any stationary point are important in -the study of thimbles. Hessians are symmetric matrices by construction. For -real actions of real variables, the study random symmetric matrices with -Gaussian entries provides insight into a wide variety of problems. In our case, -we will find the relevant ensemble is that of random symmetric matrices with -\emph{complex-normal} entries. In this section, we will introduce this -distribution, review its known properties, and derive its singular value -distribution in the large-matrix limit. +Having introduced the generic method for analytic continuation, we will now +begin dealing with the implications of actions defined in very many dimensions +with disorder. We saw in \S\ref{sec:stationary.hessian} that the singular +values of the complex hessian of the action at each stationary point are +important in the study of thimbles. Hessians are symmetric matrices by +construction. For real actions of real variables, the study of random symmetric +matrices with Gaussian entries provides insight into a wide variety of +problems. In our case, we will find the relevant ensemble is that of random +symmetric matrices with \emph{complex-normal} entries. In this section, we will +introduce this distribution, review its known properties, and derive its +singular value distribution in the large-matrix limit. The complex normal distribution with zero mean is the unique Gaussian distribution in one complex variable $Z$ whose variances are @@ -883,17 +911,21 @@ defined by }\right]^{-1}\left[\matrix{z\cr z^*}\right] \right\} \end{equation} +This is the same as writing $Z=X+iY$ and requiring that the mutual distribution +in $X$ and $Y$ be normal with $\overline{X^2}=\Gamma+\operatorname{Re}C$, +$\overline{Y^2}=\Gamma-\operatorname{Re}C$, and +$\overline{XY}=\operatorname{Im}C$. We will consider an ensemble of random $N\times N$ matrices $B=A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with variances -$\overline{|A|^2}=A_0/N$ and $\overline{A^2}=C_0/N$, and $\lambda_0$ is a -constant shift to its diagonal. The eigenvalue distribution of these matrices -is already known to take the form of an elliptical ensemble, with constant -support inside the ellipse defined by +$\overline{|A_{ij}|^2}=\Gamma_0/N$ and $\overline{A_{ij}^2}=C_0/N$, and $\lambda_0$ is a +constant shift to its diagonal. The eigenvalue distribution the matrices $A$ +is already known to take the form of an elliptical ensemble in the large-$N$ +limit, with constant support inside the ellipse defined by \begin{equation} \label{eq:ellipse} - \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{1+|C_0|/A_0}\right)^2+ - \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{1-|C_0|/A_0}\right)^2 -