Added some conclusions and full names to same bib entries.

2 files changed, 25 insertions, 2 deletions

diff --git a/stokes.bib b/stokes.bib
index 437be14..5664941 100644
--- a/stokes.bib
+++ b/stokes.bib
@@ -230,7 +230,7 @@
 }
 
 @article{Derrida_1991_The,
- author = {Derrida, B.},
+ author = {Derrida, Bernard},
  title = {The zeroes of the partition function of the random energy model},
  journal = {Physica A: Statistical Mechanics and its Applications},
  publisher = {Elsevier BV},
@@ -483,7 +483,7 @@
 }
 
 @article{Takahashi_2013_Zeros,
- author = {Takahashi, K and Obuchi, T},
+ author = {Takahashi, Kazutaka and Obuchi, Tomoyuki},
  title = {Zeros of the partition function and dynamical singularities in spin-glass systems},
  journal = {Journal of Physics: Conference Series},
  publisher = {IOP Publishing},
diff --git a/stokes.tex b/stokes.tex
index c6f603f..d09ae87 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1986,6 +1986,29 @@ points become important.
 
 \section{Conclusion}
 
+We have reviewed the Picard--Lefschetz technique for analytically continuing
+integrals and examined its applicability to the analytic continuation of phase
+space integrals over the pure $p$-spin models. The evidence suggests that
+analytic continuation is possible when weight is concentrated in gapped minima,
+who seem to avoid Stokes points, and likely impossible otherwise.
+
+This has implications for the ability to analytically continue other types of
+theories. For instance, \emph{marginal} phases of glasses, spin glasses, and
+other problems are characterized by concentration in pseudogapped minima. Based
+on the considerations of this paper, we suspect that analytic continuation is
+never possible in such a phase, as Stokes points will always proliferate among
+even the lowest minima.
+
+It is possible that a statistical theory of analytic continuation could be
+developed in order to treat these cases, whereby one computes the average or
+typical rate of Stokes points as a function of stationary point properties, and
+treats their proliferation to complex saddles as a structured diffusion
+problem. This would be a very involved calculation, involving counting exact
+classical trajectories with certain boundary conditions, but in principle it
+could be done as in \cite{Ros_2021_Dynamical}. Here the scale of the
+proliferation may save things to a degree, allowing accurate statements to be
+made about its average effects.
+
 \section*{References}
 \bibliographystyle{unsrt}
 \bibliography{stokes}