From 7bc5969c319a760c3259455e5bd24d1694ba1def Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 29 Dec 2020 19:15:47 +0100 Subject: Fixed citations. --- bezout.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'bezout.tex') diff --git a/bezout.tex b/bezout.tex index 8711dd6..d496e52 100644 --- a/bezout.tex +++ b/bezout.tex @@ -82,7 +82,7 @@ underlying simplicity that is otherwise hidden, and thus sheds light on the orig (think, for example, in the radius of convergence of a series). Deforming a real integration in $N$ variables to a surface of dimension $N$ in -the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{witten2010new,witten2011analytic}), and as a useful palliative for the sign-problem \cite{cristoforetti2012new,tanizaki2017gradient,scorzato2015lefschetz}. +the $2N$ 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, the features of landscape of the action in complex space must be understood. Such landscapes are in general not random: here we propose to follow the strategy of Computer Science of understanding the generic features of random instances, expecting that this sheds light on the practical, nonrandom problems. %Consider, for example, the @@ -460,7 +460,7 @@ dynamics, are a problem we hope to address in future work. This paper provides a first step towards the study of a complex landscape with complex variables. The next obvious one is to study the topology of the -critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{witten2010new,witten2011analytic,cristoforetti2012new,behtash2015toward,scorzato2015lefschetz}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the +critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{Witten_2010_A, Witten_2011_Analytic, Cristoforetti_2012_New, Behtash_2017_Toward, Scorzato_2016_The}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it does in the real case. -- cgit v1.2.3-54-g00ecf