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| -rw-r--r-- | bezout.tex | 18 |
1 files changed, 9 insertions, 9 deletions
@@ -86,8 +86,8 @@ 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--- 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 +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 follow the standard strategy of computer science by understanding the generic features of random instances, expecting that this sheds light on practical, nonrandom problems. @@ -106,7 +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 -thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely this. +should thus be forbidden, and the constraint surface $z^Tz=N$ accomplishes 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, @@ -116,13 +116,13 @@ z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex structure. 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 +makes the resulting configuration space 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 +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 +However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq R^2$ +on the `radius' per spin as a device to classify saddles. We shall see that this +`radius' $r$ and its upper bound $R$ are 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. @@ -155,7 +155,7 @@ 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. Based on the experience from these problems \cite{Castellani_2005_Spin-glass}, the +trick. Based on the experience from similar problems \cite{Castellani_2005_Spin-glass}, the \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is expected to be exact wherever the complexity is positive. |