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-rw-r--r--bezout.tex10
1 files changed, 6 insertions, 4 deletions
diff --git a/bezout.tex b/bezout.tex
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@@ -152,10 +152,12 @@ is shifted by $p\epsilon$.
The eigenvalue spectrum of the Hessian of the real part, or equivalently the
eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the
-singular value spectrum of $\partial\partial H$. This is a more difficult
-problem and to our knowledge a closed form for arbitrary $\kappa$ is not known.
-We have worked out an implicit form for this spectrum using the saddle point of
-a replica calculation for the Green function. blah blah blah\dots
+singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
+elements of $J$ are standard complex normal, this corresponds to a complex
+Wishart distribution. For $\kappa\neq0$ the problem changes, and to our
+knowledge a closed form is not known. We have worked out an implicit form for
+this spectrum using the saddle point of a replica calculation for the Green
+function. blah blah blah\dots
The transition from a one-cut to two-cut singular value spectrum naturally
corresponds to the origin leaving the support of the eigenvalue spectrum.