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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, 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}

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 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,
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.

\closing{Sincerely,}

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\end{letter}

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