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@@ -39,27 +39,26 @@ \opening{To the editors of Physical Review,} The subject of `complex landscapes,' which started in the spin-glass -literature, is concerned with functions (landscapes) of many variables, having -a multiplicity of minima, which are the objects of interest. Apart from its -obvious interest for glassy systems, it has found a myriad applications in many -domains: computer science, ecology, economics, biology. -\footfullcite{Mezard_2009_Information} +literature, is concerned with functions (landscapes) of many variables having +a multiplicity of minima. Apart from its obvious relevance to glassy systems, +it has found applications in many domains: computer science, ecology, +economics, and biology, to name a few. \footfullcite{Mezard_2009_Information} -In the last few years, a renewed interest has developed for landscapes for -which the variables are complex. There are a few reasons for this: {\em i)} in -computational physics, there is the main obstacle of the `sign problem', and a -strategy has emerged to attack it deforming the sampling space into complex -variables. This is a most natural and promising path, and any progress made -will have game-changing impact in solid state physics and lattice-QCD. -\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} {\em ii)} At a more basic -level, following the seminal work of E.~Witten, -\footfullcite{Witten_2010_A, Witten_2011_Analytic} there has been a flurry of activity -concerning the very definition of quantum mechanics, which requires also that -one move into the complex plane. +In the last few years, a renewed interest has developed in landscapes for +which the variables are complex. There are a couple reasons for this: in +computational physics the `sign problem' is a major obstacle, and a strategy +has emerged to attack it by deforming the sampling space into complex variables. +This is a most natural and promising path, and any progress made will have +game-changing impact in solid state physics and lattice QCD. +\footfullcite{Cristoforetti_2012_New, Scorzato_2016_The} At a more basic +level, following the seminal work of E.~Witten, \footfullcite{Witten_2010_A, +Witten_2011_Analytic} there has been a flurry of activity concerning the very +definition of quantum mechanics, which also requires that one move into the +complex plane. -In all these cases, just like in the real case, one needs to know the structure -of the `landscape.' where are the saddle points and how they are connected, -typical questions of `complexity.' However, to the best of our knowledge, +In these cases, just as in the real case, one needs to understand the structure +of the `landscape,' like the location of saddle points, how they are connected, +and typical questions of `complexity.' However, to the best of our knowledge, there are no studies extending the methods of the theory of complexity to complex variables. We believe our paper will open a field that may find numerous applications and will widen our theoretical view of complexity in |