1 files changed, 24 insertions, 0 deletions
diff --git a/marginal.tex b/marginal.tex
index fdb7377..6514eba 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -848,6 +848,12 @@ there is an energetic competition between the independent spin glass energies
 on each sphere and their tendency to align or anti-align through the
 interaction term.
 
+These models have more often been studied with random fully connected couplings
+between the spheres, for which it is possible to also use configuration spaces
+involving spheres of different sizes \cite{Subag_2021_TAP, Subag_2023_TAP,
+Bates_2022_Crisanti-Sommers, Bates_2022_Free, Huang_2023_Strong,
+Huang_2024_Optimization}.
+
 Because the energy is Gaussian, properties of the Hessian are once again
 statistically independent of those of the energy and gradient. However, unlike
 the previous example of the spherical models, the spectrum of the Hessian at
@@ -941,6 +947,24 @@ spectral density of the Hessian in these models using standard methods.
   } \label{fig:msg.marg}
 \end{figure}
 
+Fig.~\ref{fig:msg.marg} shows the examples of the Lagrange multipliers
+necessary for marginality in a set of multispherical spin glasses at various
+couplings $\epsilon$, along with some of the corresponding spectra. As
+expected, the method correctly picks out values of the Lagrange multipliers
+that result in marginal spectra.
+
+Multispherical spin glasses may be an interesting platform for testing ideas
+about which among the possible marginal minima actually attract the dynamics,
+and which do not. In the limit where $\epsilon=0$ and the configurations of the
+two spheres are independent, the minima found should be marginal on both
+sphere's energies. Just because technically on the expanded configuration space
+a deep and stable minimum on one sphere and a marginal minimum on the other is
+a marginal minimum on the whole space doesn't mean the deep and stable minimum
+is any easier to find. This intuitive idea that is precise in the zero-coupling
+limit should continue to hold at small nonzero coupling, and perhaps reveal
+something about the inherent properties of marginal minima that do not tend to be found
+by algorithms.
+
 \subsection{Random nonlinear least squares}
 \label{sec:least.squares}