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@@ -42,40 +42,42 @@ \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 +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: +defined by the energy \begin{equation} \label{eq:bare.hamiltonian} H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}, \end{equation} -where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are -real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single term -of a given $p$, this is known as the `pure $p$-spin' model, the case we shall -study here. This problem has been studied also 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 +where $J$ is a symmetric tensor whose elements are real Gaussian variables and +$z\in\mathbb R^N$ is constrained to the sphere $z^2=N$. If there is a +single term of a given $p$, this is known as the `pure $p$-spin' model, the +case we shall study here. 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 -high-energy configuration \cite{Cugliandolo_1993_Analytical}. Thanks to the -relative simplicity of the energy, all these approaches are possible -analytically in the large $N$ limit. +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. -In this paper we shall extend the study to the case where the variables are complex -$z\in\mathbb C^N$ and $J$ is a symmetric tensor whose elements are 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 becomes $z^2=N$. +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 +$\overline{J^2}=\kappa\overline{|J|^2}$ 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 +very close to ours has been proposed is the quiver Hamiltonians +\cite{Anninos_2016_Disordered} modeling black hole horizons in the zero-temperature limit. There is however a more fundamental reason for this study: we know from @@ -92,33 +94,33 @@ 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. - - - - For our model the constraint we choose $z^2=N$, -rather than $|z|^2=N$, in order to preserve the holomorphic nature of the -functions. In addition, the nonholomorphic spherical constraint has a -disturbing lack of critical points nearly everywhere, since $0=\partial^* -H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. It is enforced using the method of Lagrange multipliers: -introducing the $\epsilon\in\mathbb C$, this gives +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 easy to see that {\em for a pure $p$-spin}, at any critical point -$\epsilon=H/N$, the average energy. - -Critical points are given by the set of equations: - +For a \emph{pure} $p$-spin, $\epsilon=H/N$ -- 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 $\epsilon=0$. + +The critical points are given by the solutions to the set of equations \begin{equation} -\frac{c_p}{(p-1)!}\sum_{ i, i_2\cdots i_p}^NJ_{i, i_2\cdots i_p}z_{i_2}\cdots z_{i_p} = \epsilon z_i + \frac{p}{p!}\sum_{j_2\cdots j_p}^NJ_{ij_2\cdots j_p}z_{j_2}\cdots z_{j_p} = \epsilon z_i \end{equation} -which for given $\epsilon$ are a set of $N$ equations of degree $p-1$, to which one must add the constraint condition. -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 $p \rightarrow \infty$. - -Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical point of $\Im H_0$. 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 $\Re H_0$, which is also one of constant phase $\Im H_0=const$. +for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ are 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 $p \rightarrow \infty$. + +Since $H$ is holomorphic, a 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. Writing $z=x+iy$, $\operatorname{Re}H$ can be interpreted as a real function of $2N$ real variables. The number of critical @@ -135,7 +137,7 @@ points it has is given by the usual Kac--Rice formula: \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 -$\partial=\partial_x-i\partial_y$, one can write +$\partial=\frac12(\partial_x-i\partial_y)$, 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 @@ -158,14 +160,13 @@ that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix, 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 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 +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 $(\partial\partial -H)^\dagger\partial\partial H$. +singular values of $\partial\partial H$, 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 $N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation @@ -210,22 +211,24 @@ where \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 + \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 $\kappa\in\mathbb R$) \begin{equation} \label{eq:free.energy.a} - \begin{aligned} - 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 \\ - &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Re}\epsilon)^2 - -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Im}\epsilon)^2, - \end{aligned} + 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, \end{equation} \end{widetext} -This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot -z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter -vanishes as $a\to1$, where we should recover known results for the real +where +\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}, +\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 $\partial\partial @@ -233,28 +236,28 @@ H=\partial\partial H_0-p\epsilon I$, 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 -\eqref{eq:bare.hamiltonian} is +shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. 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} -{\bf \color{red} restricting to directions proportional to $z$, i.e. orthogonal to the constraint}, these makes its ensemble that of Gaussian complex symmetric matrices. Given its variances -$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and -$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is -constant inside the ellipse +which makes its ensemble that of Gaussian complex symmetric matrices. 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 \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 $\partial\partial H$ therefore is that of an ellipse in the complex -plane whose center lies at $-p\epsilon$. Examples of these distributions are -shown in the insets of Fig.~\ref{fig:spectra}. +spectrum of $\partial\partial H$ -- 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}. -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 +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 |