Update on Overleaf.

1 files changed, 44 insertions, 0 deletions

diff --git a/stokes.tex b/stokes.tex
index cfec1f2..06fea32 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -40,6 +40,50 @@
 
 \maketitle
 
+\section{Preamble}
+
+Complex landscapes are basically functions of many variables having many minima and, inevitably,
+many saddles of all indices, i.e. the number of unstable directions. Optimization problems 
+require us to find the deepest minima, often a difficult task.
+For example, particles with a repulsive mutual potential enclosed in a box will have many stable
+configurations, and we are asked to find the one with lowest energy.
+
+An aim of complexity theory is to be able to classify these landscapes in families having
+common properties. Two simplifications make the task potentially tractable. The first is to consider
+the limit of many variables. In the example of the particles, the limit of many particles (i.e. the thermodynamic limit)
+may be expected to bring about simplifications.
+The second simplification is of more technical nature:  we consider  functions that contain some random element to them, and we
+study the average of an ensemble. The paradigm of this is the spin-glass, where the interactions are random, and we are asked to find the
+ground state energy {\em on average over randomness}.
+
+Spin glass theory gave a surprise: random   landscapes come in two kinds:  those that
+have a `threshold level' of energy, below which there are many minima but no saddles, 
+separated by high barriers, and those that have all sorts of saddles all the way down to the lowest 
+energy levels, and local minima are separated by relatively small barriers.
+The latter are still complex, but good solutions are easier to find.
+This classification is closely related to the structure of their Replica Trick solutions.
+Armed with this solvable (random) example, it was easy to find non-random examples 
+that behave, at least approximately,  in these two ways (e.g. sphere packings and the Travelling salesman Problem,
+belong to first and second classes, respectively).
+
+What about systems whose variables are not real, but rather complex? 
+Recalling the Cauchy-Riemann conditions, we immediately see a difficulty: if our cost is, say, the real part of a function of
+$N$ complex variables, in terms of the corresponding $2N$ real variables it has only saddles of index $N$!
+Even worse: often not all saddles are equally interesting, so simply finding the lowest is not usually what we
+need to do (more about this below).
+As it turns out, there is a set of interesting questions to ask, as we describe below. For each saddle, there is a `thimble' 
+spanned by the
+lines along which the cost function decreases. The way in which these thimbles fill the complex space is crucial for many
+problems of analytic continuation, and is thus what we need to study. The central role played by saddles
+in a real landscape, the `barriers', is now played by the Stokes lines, where thimbles exchange their properties. 
+Perhaps not surprisingly, the two classes of real landscapes described above retain their role in the complex case, but now  
+the distinction is that while in the first class the Stokes lines are rare, in the second class they proliferate.
+
+In  this paper we shall start from a many-variable integral of a real function, and deform it in the many variable complex space.
+The landscape one faces is the full one in this space, and we shall see that this is an example where the proliferation -- or lack of it -- 
+of Stokes lines is the interesting quantity  in this context.
+
+
 \section{Introduction}
 
 Analytic continuation of physical theories is sometimes useful. Some theories