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| -rw-r--r-- | bezout.tex | 19 |
1 files changed, 14 insertions, 5 deletions
@@ -85,11 +85,20 @@ $2N$-dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient, -Scorzato_2016_The}. In order to do this correctly, features of landscape -of the action in complex space---such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here -we follow the standard strategy of computer science by understanding the -generic features of random instances, expecting that this sheds light on -practical, nonrandom problems. +Scorzato_2016_The}. In order to do this correctly, features of landscape of +the action in complex space---such as the relative position of saddles and the +existence of Stokes lines joining them---must be understood. This is typically +done for simple actions with few critical points, or with a target +phenomenology that possesses symmetries that restrict the set of critical +points to few candidates. Given the recent proliferation of `glassiness' in +condensed matter and high energy physics, it is inevitable that someone will +study a complex landscape with these methods, and will find old heuristic +approaches unsuitable. Such landscapes may in general not be random: here we +follow the standard strategy of computer science by understanding the generic +features of random instances of a simple case, expecting that this sheds light +on practical, nonrandom problems. While in this paper we do not address +analytic continuation of configuration space integrals, understanding the +distribution and spectra of critical points is an essential first step. Returning to our problem, the spherical constraint is enforced using the method of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our constrained |