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| -rw-r--r-- | appeal.tex | 9 | ||||
| -rw-r--r-- | bezout.tex | 32 |
2 files changed, 21 insertions, 20 deletions
@@ -52,16 +52,21 @@ referees that are beyond suspicion of incompetence or uncitedness: \begin{tabular}{ll} G Ben Arous & Courant \\ M Berry & Bristol\\ + Y. Fyodorov & King's College, London \\ Daniel Fisher& Stanford\\ - T Lubensky & Penn\\ + T Lubensky & U Penn\\ M Moore & Manchester \\ E Witten & IAS Princeton \end{tabular} -We have also added some very first results of the geometric implications for a +We have also pointed out some very first results of the geometric implications for a random complex landscape of our calculation, something we plan to expend on in a future full article. +Let us conclude by remarking that, although it is probably true that this paper will not +make more than one hundred citations next year, we are confident that it will still +be considered relevant in ten years time. + \closing{Sincerely,} @@ -69,7 +69,7 @@ $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint remains $z^Tz=N$. -The motivations for this paper are of two types. On the practical side, there +The motivations for this paper are of three types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered problems: such is the case in which the variables are \emph{phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used @@ -80,15 +80,14 @@ plane often uncovers underlying simplicity that is otherwise hidden, shedding 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 +Finally, 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, 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 +of the action in complex space--- such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here +we propose to follow standard the strategy of computer science of understanding the generic features of random instances, expecting that this sheds light on practical, nonrandom problems. @@ -107,9 +106,7 @@ holomorphic functions without any symmetry. Samples of $H_0$ nearly provide this, save for a single anomaly: the value of the energy and its gradient at any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous direction -must be neglected, and the constraint surface $z^Tz=N$ is the unique surface -whose normal is parallel to $z$ and which contains the configuration space of -the real $p$-spin model as a subspace. +thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely this. Second, taking the constraint to be the level set of a holomorphic function means the resulting configuration space is a \emph{bone fide} complex manifold, @@ -118,14 +115,13 @@ referenced above. The same cannot be said for the space defined by $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex structure. -A consequence of the constraint is that the model's configuration space is not -compact, nor is its energy bounded. This is not necessarily problematic, as many -related problems have similar properties but are concerned with subspaces on -which the energy is bounded. (In fact, identifying the appropriate subspace -often requires the study of critical points in the whole space.) Where it might -become problematic, we introduce an additional constraint that bounds the -`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$. The resulting -configuration space is a complex manifold with boundary. We shall see that the +Imposing the constraint with a holomorphic function +makes the resulting configuration space is a \emph{bone fide} complex manifold, which is, as we mentioned, the +situation we wish to model. The same cannot be said for the space defined by $z^\dagger +z=N$, which is topologically the $(2N-1)$-sphere, does not admit a complex +structure, and thus yields a trivial structure of saddles. +However, we will introduce the domains of +`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$, as a device to classify saddles. We shall see that the `radius' $r$ and its upper bound $R$ prove to be insightful knobs in our present problem, revealing structure as they are varied. Note that taking $R=1$ reduces the problem to that of the ordinary $p$-spin. @@ -159,9 +155,9 @@ usual Kac--Rice formula applied to $\operatorname{Re}H$: \end{equation} This expression is to be averaged over $J$ to give the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica -trick. In most of the parameter space that we shall study here, the +trick. Based on the experience from these problems \cite{Castellani_2005_Spin-glass}, the \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is -exact. +expected to be exact wherever the complexity is positive. The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in a manifestly complex way. With the Wirtinger derivative |