From ad1ec8fc25dc3581e6c2cfa7cc119d47e92a2185 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 9 Dec 2020 17:46:11 +0100 Subject: Added some spaces and reflowed the text to enfornce shorter lines. --- bezout.tex | 121 +++++++++++++++++++++++++++++++++++++------------------------ 1 file changed, 73 insertions(+), 48 deletions(-) diff --git a/bezout.tex b/bezout.tex index 4202e9b..5e1f44d 100644 --- a/bezout.tex +++ b/bezout.tex @@ -23,67 +23,92 @@ \date\today \begin{abstract} - We study the saddle-points of the $p$-spin model -- the best understood example of `complex (rugged) landscape' -- in the space in which all its $N$ variables are allowed to be complex. The problem becomes - a system of $N$ random equations of degree $p-1$. - We solve for quantities averaged over randomness in the $N \rightarrow \infty$ limit. - We show that the number of solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1) $\cite{Bezout_1779_Theorie}. -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. -In the case that the disorder is itself real, 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. + We study the saddle-points of the $p$-spin model -- the best understood + example of `complex (rugged) landscape' -- in the space in which all its $N$ + variables are allowed to be complex. The problem becomes a system of $N$ + random equations of degree $p-1$. We solve for quantities averaged over + randomness in the $N \rightarrow \infty$ limit. We show that the number of + solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1)$ + \cite{Bezout_1779_Theorie}. 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. In the case that the disorder is itself + real, 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. \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}) +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: \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. -Also in the Algebra \cite{Cartwright_2013_The} and Probability literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. +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. Also in the Algebra \cite{Cartwright_2013_The} and Probability +literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}. This problem 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. - -In this paper we shall extend the study to the case where $z\in\mathbb C^N$ are 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 $z^2=N$. - -The motivations for this paper are of two types. On the practical side, there are situations in which complex variables -have 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. - -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 roots. Because we are working in -complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. 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. - -This study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}. - - -Let us go back to our model. -For the constraint we choose here $z^2=N$, 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 +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. + +In this paper we shall extend the study to the case where $z\in\mathbb C^N$ +are 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 $z^2=N$. + +The motivations for this paper are of two types. On the practical side, there +are situations in which complex variables have 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. + +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 roots. Because we +are working in complex variables, and the roots are simple all the way (we +shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. 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. + +This study also provides a complement to the work on the distribution of zeroes +of random polynomials \cite{Bogomolny_1992_Distribution}. + + +Let us go back to our model. For the constraint we choose here $z^2=N$, +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 \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. +It is easy to see that {\em for a pure $p$-spin}, at any critical point +$\epsilon=H/N$, the average energy. Since $H$ is holomorphic, a point is a critical point of its real part if and -- cgit v1.2.3-70-g09d2