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Understanding the stationary points of a function can tell 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 energy landscapes in high dimensions, physical insight is often sorely lacking. Our work extends existing methods to count stationary points to a broader class of complex landscapes. Namely, this encompasses landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi that won the Nobel Prize in physics last year.

In the paper, we derive an expression for the typical number of stationary points in a class of complex random landscapes. In order to find the typical number rather than the mean number, which is biased by outliers, we 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.