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@@ -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} |