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@@ -44,7 +44,7 @@ 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 +most notably constraint satisfaction \cite{Mezard_2009_Information}. 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} @@ -72,35 +72,37 @@ 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 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 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 +problems: such is the case in which the variables 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}. A second reason +is that, as we know from experience, extending a real problem to the complex +plane often uncovers underlying simplicity that is otherwise hidden, sheding +light on the original real problem, e.g., as in the radius of convergence of a +series. + +Deforming an integral in $N$ real variables to a surface of dimension $N$ in +$2N$-dimensional complex space has turned out to be necessary for correctly +defining and analyzing path integrals with complex action (see +\cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the +sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient, +Scorzato_2016_The}. In order to do this correctly, the features of landscape +of the action in complex space---like the relative position of its +saddles---must be understood. Such landscapes are in general not random: here +we propose to follow the strategy of computer science of understanding the +generic features of random instances, expecting that this sheds light on the +practical, nonrandom problems. + +Returning to our problem, 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} - 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 $\epsilon=0$. +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 of $H$ given by the solutions to the set of equations \begin{equation} \label{eq:polynomial} @@ -108,12 +110,11 @@ The critical points are of $H$ given by the solutions to the set of equations = p\epsilon z_i \end{equation} 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 +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 $p\to\infty$. -We see from \eqref{eq:polynomial} that at any critical point, $\epsilon=H/N$, the average energy. +and $p\to\infty$. We see from \eqref{eq:polynomial} that at any critical +point, $\epsilon=H/N$, the average energy. 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 @@ -444,18 +445,23 @@ 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 gradient 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. + This paper provides a first step towards the study of a complex landscape with + complex variables. The next obvious one is to study the topology of the + critical points, the sets reached following gradient descent (the + Lefschetz thimbles), and ascent (the anti-thimbles) \cite{Witten_2010_A, + Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, + Scorzato_2016_The}, which act as constant-phase integrating `contours.' + Locating and counting the saddles that are joined by gradient lines---the + Stokes points, which play an important role in the theory---is also well within + reach of the present-day spin-glass literature techniques. 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} We wish to thank Alexander Altland, Satya Majumdar and Gregory Schehr for a useful suggestions. JK-D and JK are supported by the Simons Foundation Grant No.~454943. \end{acknowledgments} -\bibliographystyle{apsrev4-2} \bibliography{bezout} \end{document} |