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@@ -175,6 +175,20 @@ doi = {10.1103/physrevlett.71.173} } +@article{Dyson_1962_A, + author = {Dyson, Freeman J.}, + title = {A Brownian-Motion Model for the Eigenvalues of a Random Matrix}, + journal = {Journal of Mathematical Physics}, + publisher = {AIP Publishing}, + year = {1962}, + month = {11}, + number = {6}, + volume = {3}, + pages = {1191--1198}, + url = {https://doi.org/10.1063%2F1.1703862}, + doi = {10.1063/1.1703862} +} + @article{Fyodorov_2004_Complexity, author = {Fyodorov, Yan V.}, title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices}, @@ -29,100 +29,98 @@ of degree $p-1$. We solve for $\overline{\mathcal{N}}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$. - The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b - B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$ are GOE matrices and $a-d$ - real. Its spectrum has a transition from one-cut to two-cut that generalizes - the notion of `threshold level' that is well-known in the real problem. The - results from the real problem are recovered in the limit of real disorder. In - this case, only the square-root of the total number solutions are real. In - terms of real and imaginary parts of the energy, the solutions are divided in - sectors where the saddles have different topological properties. + The Hessian of each saddle is given by a random matrix of the form $C^\dagger + C$, where $C$ is a complex Gaussian matrix with a shift to its diagonal. Its + spectrum has a transition where a gap develops that generalizes the notion of + `threshold level' well-known in the real problem. The results from the real + problem are recovered in the limit of real parameters. In this case, only the + square-root of the total number of solutions are real. In terms of the + complex energy, the solutions are divided into sectors where the saddles have + different topological properties. \end{abstract} \maketitle -Spin-glasses have long been considered the paradigm of `complex landscapes' of -many variables, a subject that includes neural networks and optimization -problems, most notably constraint satisfaction ones. The most tractable -family of these are the mean-field spherical $p$-spin models -\cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass}) -defined by the energy +Spin-glasses have long been considered the paradigm of many variable `complex +landscapes,' a subject that includes neural networks and optimization problems, +most notably constraint satisfaction. The most tractable family of these +are the mean-field spherical $p$-spin models \cite{Crisanti_1992_The} (for a +review see \cite{Castellani_2005_Spin-glass}) defined by the energy \begin{equation} \label{eq:bare.hamiltonian} H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, \end{equation} where $J$ is a symmetric tensor whose elements are real Gaussian variables and -$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been studied in the algebra -\cite{Cartwright_2013_The} and probability literature +$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. This problem has been +studied in the algebra \cite{Cartwright_2013_The} and probability literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. It has been attacked from several angles: the replica trick to compute the Boltzmann--Gibbs distribution \cite{Crisanti_1992_The}, a Kac--Rice \cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the Fadeev--Popov integral) to compute the number of saddle-points of the energy function \cite{Crisanti_1995_Thouless-Anderson-Palmer}, and the -gradient-descent -- or more generally Langevin -- dynamics staring from a +gradient-descent---or more generally Langevin---dynamics staring from a high-energy configuration \cite{Cugliandolo_1993_Analytical}. Thanks to the simplicity of the energy, all these approaches yield analytic results in the -large $N$ limit. +large-$N$ limit. -In this paper we extend the study to the case where the variables are complex: -we shall take $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements -are \emph{complex} normal, with $\overline{|J|^2}=p!/2N^{p-1}$ and +In this paper we extend the study to complex variables: we shall take +$z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are +\emph{complex} normal, with $\overline{|J|^2}=p!/2N^{p-1}$ and $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint remains $z^2=N$. The motivations for this paper are of two types. On the practical side, there -are indeed situations in which complex variables in a disorder problem appear -naturally: such is the case in which they are {\em phases}, as in random laser -problems \cite{Antenucci_2015_Complex}. Another problem where a Hamiltonian -very close to ours has been proposed is the quiver Hamiltonians -\cite{Anninos_2016_Disordered} modeling black hole horizons in the -zero-temperature limit. +are indeed situations in which complex variables appear naturally in disordered +problems: such is the case in which they are \emph{phases}, as in random laser +problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used to model +black hole horizons in the zero-temperature limit---also have a Hamiltonian +very close to ours \cite{Anninos_2016_Disordered}. There is however a more fundamental reason for this study: we know from -experience that extending a problem to the complex plane often uncovers an -underlying simplicity that is hidden in the purely real case. Consider, for -example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ -and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from -$\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ -is a polynomial of degree $p$ chosen to have simple, known saddles. Because we -are working in complex variables, and the saddles are simple all the way (we -shall confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, while with -real variables minima of functions appear and disappear, and this procedure is -not possible. The same idea may be implemented by performing diffusion in the -$J$'s, and following the roots, in complete analogy with Dyson's stochastic -dynamics. +experience that extending a real problem to the complex plane often uncovers +underlying simplicity that is otherwise hidden. Consider, for example, the +procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying +$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ +to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a +polynomial of degree $p$ chosen to have simple, known saddles. Because we are +working in complex variables, and the saddles are simple all the way (we shall +confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, +while with real variables minima of functions appear and disappear, and this +procedure is not possible. The same idea may be implemented by performing +diffusion in the $J$s and following the roots, in complete analogy with Dyson's +stochastic dynamics \cite{Dyson_1962_A}. The spherical constraint is enforced using the method of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). \end{equation} -It is easily shown that $\epsilon=H/N$ -- the average energy -- at any -critical point. We choose to constrain our model by $z^2=N$ rather than -$|z|^2=N$ in order to preserve the holomorphic nature of $H$. In addition, the -nonholomorphic spherical constraint has a disturbing lack of critical points -nearly everywhere: if $H$ was so constrained, then $0=\partial^* H=-p\epsilon -z$ would only be satisfied for $\epsilon=0$. +At any critical point, $\epsilon=H/N$, the average energy. We choose to +constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to preserve the +analyticity of $H$. The nonholomorphic constraint also has a disturbing lack of +critical points nearly everywhere: if $H$ were so constrained, then +$0=\partial^* H=-p\epsilon z$ would only be satisfied for $\epsilon=0$. -The critical points are given by the solutions to the set of equations +The critical points are of $H$ given by the solutions to the set of equations \begin{equation} - \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}} = \epsilon z_i + \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}} + = p\epsilon z_i \end{equation} -for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ is a set of $N$ +for all $i=\{1,\ldots,N\}$, which for fixed $\epsilon$ is a set of $N$ equations of degree $p-1$, to which one must add the constraint. In this sense this study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ -and $p \rightarrow \infty$. +and $p\to\infty$. -Since $H$ is holomorphic, a critical point of $\operatorname{Re}H$ is also a +Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a critical point of $\operatorname{Im}H$. The number of critical points of $H$ is -therefore the number of critical points of $\operatorname{Re}H$. From each -critical point emerges a gradient line of $\operatorname{Re}H$, which is also -one of constant $\operatorname{Im}H$ and therefore constant phase. +therefore the same as that of $\operatorname{Re}H$. From each critical point +emerges a gradient line of $\operatorname{Re}H$, which is also one of constant +$\operatorname{Im}H$ and therefore constant phase. -Writing $z=x+iy$, $\operatorname{Re}H$ can be -interpreted as a real function of $2N$ real variables. The number of critical -points it has is given by the usual Kac--Rice formula: +Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function +of $2N$ real variables. Its number of critical points is given by the usual +Kac--Rice formula: \begin{equation} \label{eq:real.kac-rice} \begin{aligned} \mathcal N_J(\kappa,\epsilon) @@ -133,12 +131,11 @@ points it has is given by the usual Kac--Rice formula: \end{bmatrix}\right|. \end{aligned} \end{equation} -The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in -a manifestly complex way. Using the Wirtinger derivative -$\partial=\frac12(\partial_x-i\partial_y)$, one can write -$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and -$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying -these transformations through, we have +The Cauchy--Riemann equations may be used to write this in a manifestly complex +way. With the Wirtinger derivative $\partial=\frac12(\partial_x-i\partial_y)$, +one can write $\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and +$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these +transformations through, we have \begin{equation} \label{eq:complex.kac-rice} \begin{aligned} &\mathcal N_J(\kappa,\epsilon) @@ -159,46 +156,35 @@ or the norm squared of that of an $N\times N$ complex symmetric matrix. These equivalences belie a deeper connection between the spectra of the corresponding matrices. Each positive eigenvalue of the real matrix has a -negative partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is -an eigenvalue of the Hermitian matrix and $|\lambda|$ is a \emph{singular -value} of the complex symmetric matrix. The distribution of positive -eigenvalues of the Hessian is therefore the same as the distribution of -singular values of $\partial\partial H$, the distribution of square-rooted -eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$. +negative partner. For each such pair $\pm\lambda$, $\lambda^2$ is an eigenvalue +of the Hermitian matrix and $|\lambda|$ is a \emph{singular value} of the +complex symmetric matrix. The distribution of positive eigenvalues of the +Hessian is therefore the same as the distribution of singular values of +$\partial\partial H$, or the distribution of square-rooted eigenvalues of +$(\partial\partial H)^\dagger\partial\partial H$. -The expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as -$N \Sigma= \overline{\log\mathcal N} = \int dJ \, \log \mathcal N_J$, a calculation -that involves the replica trick. In most the parameter-space that we shall -study here, the {\em annealed approximation} $N \Sigma \sim \log \overline{ -\mathcal N} = \log\int dJ \, \mathcal N_J$ is exact. +The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give +the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \, +\log \mathcal N_J$, a calculation that involves the replica trick. In most the +parameter-space that we shall study here, the \emph{annealed approximation} $N +\Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is +exact. -A useful property of the Gaussian distributions is that gradient and Hessian -for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics, -Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the -Hessians as independent. We compute each by taking the saddle point. The -$\delta$-functions are converted to exponentials by the introduction of -auxiliary fields $\hat z=\hat x+i\hat y$. The average over $J$ can then be -performed. A generalized Hubbard--Stratonovich then allows a change of -variables from the $4N$ original and auxiliary fields to eight bilinears -defined by -\begin{equation} - \begin{aligned} - Na=z^*\cdot z - && - N\hat c=\hat z\cdot\hat z - && - Nb=\hat z^*\cdot z \\ - N\hat a=\hat z^*\cdot\hat z - && - Nd=\hat z\cdot z - \end{aligned} -\end{equation} -and their conjugates. The result is, to leading order in $N$, +A useful property of the Gaussian $J$ is that gradient and Hessian at fixed +$\epsilon$ are statistically independent \cite{Bray_2007_Statistics, +Fyodorov_2004_Complexity}, so that the $\delta$-functions and the Hessian may +be averaged independently. The $\delta$-functions are converted to exponentials +by the introduction of auxiliary fields $\hat z=\hat x+i\hat y$. The average +of those factors over $J$ can then be performed. A generalized +Hubbard--Stratonovich allows a change of variables from the $4N$ original +and auxiliary fields to eight bilinears defined by $Na=|z|^2$, $N\hat a=|\hat +z|^2$, $N\hat c=\hat z^2$, $Nb=\hat z^*z$, and $Nd=\hat zz$ (and their +conjugates). The result, to leading order in $N$, is \begin{equation} \label{eq:saddle} - \overline{\mathcal N_J}(\kappa,\epsilon) + \overline{\mathcal N}(\kappa,\epsilon) = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)}, \end{equation} -where +where the argument of the exponential is \begin{widetext} \begin{equation} f=2+\frac12\log\det\frac12\begin{bmatrix} @@ -207,64 +193,61 @@ where d & b^* & \hat c & \hat a \\ b & d^* & \hat a & \hat c^* \end{bmatrix} - +\operatorname{Re}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\} +\int d\lambda\,\rho(\lambda)\log|\lambda|^2 + +\operatorname{Re}\left\{ + \frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b + \right\}. \nonumber % He's too big! \end{equation} - where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of - $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in - $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with - $\kappa\in\mathbb R$) + The integral of the distribution $\rho$ of eigenvalues of $\partial\partial + H$ comes from the Hessian and is dependant on $a$ alone. This function has a + maximum in $\hat a$, $b$, $\hat c$, and $d$ at which its value is \begin{equation} \label{eq:free.energy.a} - f(a)=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 - -C_+(a)(\operatorname{Re}\epsilon)^2-C_-(a)(\operatorname{Im}\epsilon)^2, + f(a)=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-|\kappa|^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 + -2C_+[\operatorname{Re}(\epsilon e^{-i\theta})]^2-2C_-[\operatorname{Im}(\epsilon e^{-i\theta})]^2, \end{equation} \end{widetext} -where +where $\theta=\frac12\arg\kappa$ and \begin{equation} - C_{\pm}(a)=\frac{a^p(1+p(a^2-1))\mp a^2\kappa}{a^{2p}\pm a^p(a^2-1)(p-1)-a^2\kappa^2}, + C_{\pm}=\frac{a^p(1+p(a^2-1))\mp a^2|\kappa|}{a^{2p}\pm(p-1)a^p(a^2-1)|\kappa|-a^2|\kappa|^2}. \end{equation} This leaves a single parameter, $a$, which dictates the magnitude of $|z|^2$, or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes as $a\to1$, where (as we shall see) one recovers known results for the real $p$-spin. -The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial -H=\partial\partial H_0-p\epsilon I$, or the Hessian of -\eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The -eigenvalue distribution $\rho$ of the constrained Hessian is therefore related -to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar -shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of the unconstrained Hamiltonian is +The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to +the unconstrained Hessian with a constant added to its diagonal. The eigenvalue +distribution $\rho$ is therefore related to the unconstrained distribution +$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The +Hessian of the unconstrained Hamiltonian is \begin{equation} \label{eq:bare.hessian} \partial_i\partial_jH_0 =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, \end{equation} -which makes its ensemble that of Gaussian complex symmetric matrices when the +which makes its ensemble that of Gaussian complex symmetric matrices, when the direction along the constraint is neglected. Given its variances $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and -$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, its distribution of -eigenvalues $\rho_0(\lambda)$ is constant inside the ellipse +$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is +constant inside the ellipse \begin{equation} \label{eq:ellipse} \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+ \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2 <\frac{p(p-1)}{2a^{p-2}} \end{equation} where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue -spectrum of $\partial\partial H$ -- the constrained Hessian -- is therefore -that of the same ellipse whose center lies at $-p\epsilon$. -Examples of these distributions are shown in the insets of -Fig.~\ref{fig:spectra}. +spectrum of $\partial\partial H$ is therefore constant inside the same ellipse +translated so that its center lies at $-p\epsilon$. Examples of these +distributions are shown in the insets of Fig.~\ref{fig:spectra}. -The eigenvalue spectrum of the Hessian of the real part is the one we need for -our Kac--Rice formula. It is different from the spectrum $\partial\partial H$, -but rather equivalent to the square-root eigenvalue spectrum of -$(\partial\partial H)^\dagger\partial\partial H$, in other words, the singular -value spectrum of $\partial\partial H$. When $\kappa=0$ and the elements of $J$ -are standard complex normal, this corresponds to a complex Wishart -distribution. For $\kappa\neq0$ the problem changes, and to our knowledge a -closed form is not in the literature. We have worked out an implicit form for -this spectrum using the saddle point of a replica symmetric calculation for the -Green function. +The eigenvalue spectrum of the Hessian of the real part is different from the +spectrum $\rho(\lambda)$ of $\partial\partial H$, but rather equivalent to the +square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$; +in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial +H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this +is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and +to our knowledge a closed form is not in the literature. We have worked out an +implicit form for this spectrum using the replica method. \begin{figure}[htpb] \centering @@ -288,8 +271,8 @@ Green function. } \label{fig:spectra} \end{figure} -Introducing replicas to bring the partition function to -the numerator of the Green function \cite{Livan_2018_Introduction} gives +Introducing replicas to bring the partition function into the numerator of the +Green function \cite{Livan_2018_Introduction} gives \begin{widetext} \begin{equation} \label{eq:green.replicas} G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} @@ -297,15 +280,15 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right) \right] - \right\} + \right\}, \end{equation} - with sums taken over repeated latin indices. - The average can then be made over $J$ and Hubbard--Stratonovich used to change - variables to replica matrices + with sums taken over repeated Latin indices. The average is then made over + $J$ and Hubbard--Stratonovich is used to change variables to the replica matrices $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^*\cdot\zeta^{(\beta)}$ and - $N\chi_{\alpha\beta}=\zeta^{(\alpha)}\cdot\zeta^{(\beta)}$ and a series of replica - vectors. Taking the replica-symmetric ansatz leaves all off-diagonal elements - and vectors zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$, + $N\chi_{\alpha\beta}=\zeta^{(\alpha)}\cdot\zeta^{(\beta)}$ and a series of + replica vectors. The replica-symmetric ansatz leaves all off-diagonal + elements and vectors zero, and + $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$, $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is \begin{equation}\label{eq:green.saddle} \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0 @@ -317,8 +300,8 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives \end{equation} \end{widetext} The argument of the exponential has several saddles. The solutions $\alpha_0$ -are the roots of a sixth-order polynomial, but the root with the -smallest value of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct +are the roots of a sixth-order polynomial, and the root with the smallest value +of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct solution. A detailed analysis of the saddle point integration is needed to understand why this is so. Given such $\alpha_0$, the density of singular values follows from the jump across the cut, or @@ -335,7 +318,7 @@ Weyl's theorem requires that the product over the norm of all eigenvalues must not be greater than the product over all singular values \cite{Weyl_1912_Das}. Therefore, the absence of zero eigenvalues implies the absence of zero singular values. The determination of the threshold energy -- the energy at which the -distribution of singular values becomes gapped -- is therefore reduced to a +distribution of singular values becomes gapped -- is then reduced to a geometry problem, and yields \begin{equation} \label{eq:threshold.energy} |\epsilon_{\mathrm{th}}|^2 @@ -344,79 +327,59 @@ geometry problem, and yields \end{equation} for $\delta=\kappa a^{-(p-2)}$. -With knowledge of this distribution, the integral in \eqref{eq:free.energy.a} -may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$, -the resulting expression is always maximized for $a=\infty$. Taking this saddle -gives +Given $\rho$, the integral in \eqref{eq:free.energy.a} may be preformed for +arbitrary $a$. The resulting expression is maximized for $a=\infty$ for all +values of $\kappa$ and $\epsilon$. Taking this saddle gives \begin{equation} \label{eq:bezout} \log\overline{\mathcal N}(\kappa,\epsilon) - ={N\log(p-1)} + =N\log(p-1). \end{equation} This is precisely the Bézout bound, the maximum number of solutions that $N$ equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is -gained by looking at the count as a function of $a$, defined by -\begin{equation} \label{eq:count.def.marginal} - {\mathcal N}(\kappa,\epsilon,a) - ={\mathcal N}(\kappa,\epsilon/ \sum_i y_i^2<Na) -\end{equation} -and likewise the $a$-dependant complexity -$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}_a(\kappa,\epsilon,a)$ -the large-$N$ limit, the $a$-dependant expression may be considered the -cumulative number of critical points up to the value $a$. +gained by looking at the count as a function of $a$, defined by $\overline{\mathcal +N}(\kappa,\epsilon,a)=e^{Nf(a)}$. In the large-$N$ limit, this is the +cumulative number of critical points, or the number of critical points $z$ for +which $|z|^2\leq a$. We likewise define the $a$-dependant complexity +$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$ + +\begin{figure}[htpb] + \centering + \includegraphics{fig/complexity.pdf} + \caption{ + The complexity of the 3-spin model at $\epsilon=0$ as a function of + $a=|z|^2=1+y^2$ at several values of $\kappa$. The dashed line shows + $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. + } \label{fig:complexity} +\end{figure} -The integral in \eqref{eq:free.energy.a} can only be performed explicitly for -certain ellipse geometries. One of these is at $\epsilon=0$ any values of -$\kappa$ and $a$, which yields the $a$-dependent complexity +Everything is analytically tractable for $\epsilon=0$, giving \begin{equation} \label{eq:complexity.zero.energy} \Sigma(\kappa,0,a) =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2a^{-2(p-1)}}{1-a^{-2}}\right). \end{equation} Notice that the limit of this expression as $a\to\infty$ corresponds with -\eqref{eq:bezout}, as expected. Equation \eqref{eq:complexity.zero.energy} is -plotted as a function of $a$ for several values of $\kappa$ in -Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to -negative infinity as $a\to1$, i.e., as the spins are restricted to the reals. -This is natural, given that the $y$ contribution to the volume shrinks to zero -as that of an $N$-dimensional sphere $\sim(a-1)^N$. However, when the result -is analytically continued to $\kappa=1$ (which corresponds to real $J$) -something novel occurs: the complexity has a finite value at $a=1$. Since the -$a$-dependence gives a cumulative count, this implies a $\delta$-function -density of critical points along the line $y=0$. The number of critical points -contained within is +\eqref{eq:bezout}, as expected. This is plotted as a function of $a$ for +several values of $\kappa$ in Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, +the complexity goes to negative infinity as $a\to1$, i.e., as the spins are +restricted to the reals. This is natural, given that the $y$ contribution to +the volume shrinks to zero as that of an $N$-dimensional sphere with volume +$\sim(a-1)^N$. However, when the result is analytically continued to +$\kappa=1$ (which corresponds to real $J$) something novel occurs: the +complexity has a finite value at $a=1$. Since the $a$-dependence gives a +cumulative count, this implies a $\delta$-function density of critical points +along the line $y=0$. The number of critical points contained within is \begin{equation} \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a) = \frac12N\log(p-1), \end{equation} half of \eqref{eq:bezout} and corresponding precisely to the number of critical -points of the real $p$-spin model. (note the role of conjugation -symmetry, already underlined in \cite{Bogomolny_1992_Distribution}). The full +points of the real $p$-spin model (note the role of conjugation symmetry, +already underlined in \cite{Bogomolny_1992_Distribution}). The full $\epsilon$-dependence of the real $p$-spin is recovered by this limit as $\epsilon$ is varied. \begin{figure}[htpb] \centering - \includegraphics{fig/complexity.pdf} - \caption{ - The complexity of the 3-spin model at $\epsilon=0$ as a function of - $a$ at several values of $\kappa$. The dashed line shows - $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. - } \label{fig:complexity} -\end{figure} - -These qualitative features carry over to nonzero $\epsilon$. In -Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ -close to one for which there are no solutions. For the case $\kappa=1$ -- the -analytic continuation to the usual real computation -- the situation is more -interesting. In the range of energies where there are real solutions this gap -closes, and this is only possible if the density of solutions diverges at -$a=1$. Another remarkable feature of the limit $\kappa=1$ is that there is -still a gap without solutions around `deep' real energies where there is no -real solution. A moment's thought tells us that this is a necessity: otherwise -a small perturbation of the $J$'s could produce a real, unusually deep solution -for the real problem, in a region where we expect this not to happen. - -\begin{figure}[htpb] - \centering \includegraphics{fig/desert.pdf} \caption{ The minimum value of $a$ for which the complexity is positive as a function @@ -425,20 +388,17 @@ for the real problem, in a region where we expect this not to happen. } \label{fig:desert} \end{figure} -The relationship between the threshold and ground -- or more generally, extremal -- state energies is richer than -in the real case. In Fig.~\ref{fig:eggs} these are shown in the -complex-$\epsilon$ plane for several examples. Depending on the parameters, the -threshold line always come at smaller magnitude than the ground state, or always -come at larger magnitude than the ground state, or cross as a -function of complex argument. For sufficiently large $a$ the threshold always -comes at larger magnitude than the ground state. If this were to happen in the -real case, it would likely imply our replica symmetric computation is unstable, -as having the ground state above the threshold would imply a ground state -Hessian with many negative eigenvalues, a contradiction with the notion of a -ground state. However, this is not an obvious contradiction in the complex case. -The relationship between the threshold, i.e., -where the gap appears, and the dynamics of, e.g., a minimization algorithm or -physical dynamics, are a problem we hope to address in future work. +These qualitative features carry over to nonzero $\epsilon$. In +Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ +close to one for which there are no solutions. When $\kappa=1$---the analytic +continuation to the real computation---the situation is more interesting. In +the range of energies where there are real solutions this gap closes, which is +only possible if the density of solutions diverges at $a=1$. Another +remarkable feature of this limit is that there is still a gap without solutions +around `deep' real energies where there is no real solution. A moment's thought +tells us that this is a necessity: otherwise a small perturbation of the $J$s +could produce an unusually deep solution to the real problem, in a region where +this should not happen. \begin{figure}[htpb] \centering @@ -452,15 +412,30 @@ physical dynamics, are a problem we hope to address in future work. Energies at which states exist (green shaded region) and threshold energies (black solid line) for the 3-spin model with $\kappa=\frac34e^{-i3\pi/4}$ and (a) $a=2$, (b) $a=1.325$, (c) $a=1.125$, - and (d) $a=1$. No shaded region is shown in (d) because no states exist an + and (d) $a=1$. No shaded region is shown in (d) because no states exist at any energy. } \label{fig:eggs} \end{figure} -This paper provides a first step for the study of a complex landscape with complex variables. The next obvious one -is to study the topology of the critical points and the lines of constant phase. -We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it -does in the real case. +The relationship between the threshold and ground, or extremal, state energies +is richer than in the real case. In Fig.~\ref{fig:eggs} these are shown in the +complex-$\epsilon$ plane for several examples. Depending on the parameters, the +threshold might always come at smaller magnitude than the extremal state, or +always come at larger magnitude, or cross as a function of complex argument. +For sufficiently large $a$ the threshold always comes at larger magnitude than +the extremal state. If this were to happen in the real case, it would likely +imply our replica symmetric computation is unstable, since having a ground +state above the threshold implies a ground state Hessian with many negative +eigenvalues, a contradiction. However, this is not an obvious contradiction in +the complex case. The relationship between the threshold, i.e., where the gap +appears, and the dynamics of, e.g., a minimization algorithm or physical +dynamics, are a problem we hope to address in future work. + +This paper provides a first step for the study of a complex landscape with +complex variables. The next obvious one is to study the topology of the +critical points and the lines of constant phase. We anticipate that the +threshold level, where the system develops a mid-spectrum gap, will play a +crucial role as it does in the real case. \begin{acknowledgments} JK-D and JK are supported by the Simons Foundation Grant No.~454943. |