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diff --git a/marginal.tex b/marginal.tex index 190c3be..3eeed6b 100644 --- a/marginal.tex +++ b/marginal.tex @@ -650,9 +650,9 @@ The ordinary Kac--Rice calculation involves many moving parts, and this method for incorporating marginality adds even more. It is therefore convenient to introduce compact and simplifying notation through a superspace representation. The use of superspace in the Kac--Rice calculation is well established, as well -as the deep connections with BRST symmetry that is implied. +as the deep connections with BRST symmetry that is implied \cite{Annibale_2003_Supersymmetric, Annibale_2003_The, Annibale_2004_Coexistence}. Appendix~\ref{sec:superspace} introduces the notation and methods of -superspace. Here we describe how it can be used to simplify the complexity +superspace algebra. Here we describe how it can be used to simplify the complexity calculation for marginal minima. We consider the $\mathbb R^{N|4}$ superspace whose Grassmann indices are @@ -687,7 +687,13 @@ Here we have also defined the operator (1-\hat\beta\bar\theta_1\theta_1) -\delta^{\alpha1}\hat\lambda-\beta \end{equation} -which encodes various aspects of the complexity problem, and the measures +which encodes various aspects of the complexity problem. When the Lagrangian is +expanded in a series with respect to the Grassmann indices and the definition +of $B$ inserted, the result of the Grassmann integrals produces exactly the +content of the integrand in \eqref{eq:min.complexity.expanded} with the +substitutions \eqref{eq:delta.grad}, \eqref{eq:delta.energy}, +\eqref{eq:delta.eigen}, and \eqref{eq:determinant} of the Dirac $\delta$ +functions and the determinant made. The new measures \begin{align} d\pmb\phi_a^\alpha &=\left[ @@ -702,7 +708,7 @@ which encodes various aspects of the complexity problem, and the measures d\pmb\omega&=\bigg(\prod_{i=1}^rd\omega_i\bigg) \,\delta\bigg(N\mu-\sum_i^r\omega_i\operatorname{Tr}\partial\partial g_i\bigg) \end{align} -that collect the individual measures of the various fields embedded in the superfield. +collect the individual measures of the various fields embedded in the superfield, along with their constraints. \end{widetext} With this way of writing the replicated count, the problem of marginal complexity temporarily takes the schematic form of an equilibrium calculation @@ -710,6 +716,7 @@ with configurations $\pmb\phi$, inverse temperature $B$, and energy $L$. This makes the intermediate pieces of the calculation dramatically simpler. Of course the intricacies of the underlying problem are not banished: near the end of the calculation, terms involving the superspace must be expanded. +We will make use of this representation to simplify the analysis of the marginal complexity when analyzing random sums of squares in Section \ref{sec:least.squares}. \section{Examples} \label{sec:examples} @@ -1251,7 +1258,7 @@ subspace and a stable minimum on the other. \textbf{(b)}~Spectra corresponding to the parameters $\omega_1$ and $\omega_2$ marked by the circles in panel (a). \textbf{(c)}~The complexity of marginal minima in a multispherical model with - $f_i(q)=q^{p_i}/[p_i(p_i-1)]$ for $p_1=3$ and $p_2=4$ for a variety of + $f_1(q)=\frac16q^3$ and $f_2(q)=\frac1{12}q^4$ for a variety of $\epsilon$. Since $f_1''(1)=f_2''(1)=1$, the marginal values correspond precisely to those in (a--b). } \label{fig:msg.marg} |