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\signature{
  \vspace{-6\medskipamount}
  \smallskip
  Jaron Kent-Dobias \& Jorge Kurchan
}

\address{
  Laboratoire de Physique\\
  Ecole Normale Sup\'erieure\\
  24 rue Lhomond\\ 
  75005 Paris
}

\begin{document}
\begin{letter}{
  Editorial Office\\
  Physical Review Letters\\
  1 Research Road\\
  Ridge, NY 11961
}

\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. 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}

Recently, interest has developed in landscapes for which the variables are
complex. There are several 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 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
general. Our manuscript has been amended to emphasize these important connections with other areas of physics.

\closing{Sincerely,}

\vspace{1em}

\end{letter}

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