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Understanding the stationary points of a function tells you a lot about the function. From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. For complex or rugged energy landscapes of many variables, insight is often sorely lacking, hence the very intense interdisciplinary activity on the subject over the last three decades. Our work extends existing methods to count stationary points to a substantially broader class of complex landscapes. Namely, this encompasses all complex landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi recognized by the Nobel Prize in physics last year.

In the paper, we derive an expression for the typical number of stationary points in these landscapes. In order to find the typical number, one must use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes.

A correct accounting of stationary points for these and other complex landscapes promises to yield important insight into physics in many disciplines where such landscapes appear, from the condensed matter of glasses to the performance of machine learning algorithms.