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@@ -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 |