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A common measure of a function's complexity is the count of its stationarypoints. For complicated functions, this count grows exponentially with thevolume and dimension of their domain. In practice, the count is averaged over aclass of functions (the annealed average), but the large numbers involved canproduce averages biased by extremely rare samples. Typical counts are reliablyfound by taking the average of the logarithm (the quenched average), which ismore difficult and not often done in practice. When most stationary points areuncorrelated with each other, quenched and anneals averages are equal.Equilibrium heuristics can guarantee when most of the lowest minima will beuncorrelated. We show that these equilibrium heuristics cannot be used to drawconclusions about other minima and saddles by producing examples amongGaussian-correlated functions on the hypersphere where the count of certainsaddles and minima has different quenched and annealed averages, despite beingguaranteed `safe' in the equilibrium setting. We determine conditions for theemergence of nontrivial correlations between saddles, and discuss theimplications for the geometry of those functions and what out-of-equilibriumsettings might be affected.

When is the average number of saddle points typical?