From d9701957da92a97eda685ac44864b12d665a285d Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Tue, 29 Dec 2020 17:20:01 +0000 Subject: Update on Overleaf. --- bezout.tex | 50 ++++++++++++++++++++++++++++++++------------------ 1 file changed, 32 insertions(+), 18 deletions(-) (limited to 'bezout.tex') diff --git a/bezout.tex b/bezout.tex index 9e701f5..8711dd6 100644 --- a/bezout.tex +++ b/bezout.tex @@ -72,26 +72,40 @@ 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: 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}. - -There is however a more fundamental reason for this study: we know from +A second reason is that 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: +underlying simplicity that is otherwise hidden, and thus sheds light on the original real problem +(think, for example, in the radius of convergence of a series). + +Deforming a real integration in $N$ variables to a surface of dimension $N$ in +the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{witten2010new,witten2011analytic}), and as a useful palliative for the sign-problem \cite{cristoforetti2012new,tanizaki2017gradient,scorzato2015lefschetz}. +In order to do this correctly, the features of landscape of the action in complex space 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. + +%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 $\lambda=0$ +%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 +% +%$\lambda H_{00} + (1-\lambda) H_0$ evolving adiabatically from $\lambda=1$ to $\lambda=0$, as +%is familiar from quantum annealing. The $H_{00}$ is a polynomial +%of degree N 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}. + +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). @@ -444,9 +458,9 @@ 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 + 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 and gradient lines of constant phase. We anticipate that the +critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{witten2010new,witten2011analytic,cristoforetti2012new,behtash2015toward,scorzato2015lefschetz}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that 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. -- cgit v1.2.3-54-g00ecf