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| -rw-r--r-- | bezout.tex | 30 |
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@@ -80,23 +80,25 @@ plane often uncovers underlying simplicity that is otherwise hidden, shedding light on the original real problem, e.g., as in the radius of convergence of a series. -Finally, deforming an integral in $N$ real variables to a surface of dimension $N$ in -$2N$-dimensional complex space has turned out to be necessary for correctly -defining and analyzing path integrals with complex action (see +Finally, deforming an integral in $N$ real variables to a surface of dimension +$N$ in $2N$-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 the action's landscape in complex space---such as the relative position of saddles and the +Scorzato_2016_The}. In order to do this correctly, features of the action's +landscape 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 saddles, or for a target -phenomenology with symmetries that restrict the set of saddles to few candidates. Given the recent proliferation of `glassiness' in -condensed matter and high energy physics, it is inevitable that someone will -want to apply these methods to a system with a complex landscape, and will find -they cannot use approaches that rely on such assumptions. Their landscape may 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 yet address -analytic continuation of integrals, understanding the distribution and spectra -of critical points is an essential first step. +done for simple actions with few saddles, or for a target phenomenology with +symmetries that restrict the set of saddles to few candidates. Given the recent +proliferation of `glassiness' in condensed matter and high energy physics, it +is inevitable that someone will want to apply these methods to a system with a +complex landscape, and will find they cannot use approaches that rely on such +assumptions. Their landscape may 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 yet address analytic +continuation of 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 |