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Oscillatory critical amplitudes have been repeatedly observed in hierarchicalmodels and, in the cases that have been taken into consideration, theseoscillations are so small to be hardly detectable. Hierarchical models aretightly related to iteration of maps and, in fact, very similar phenomena havebeen repeatedly reported in many fields of mathematics, like combinatorialevaluations and discrete branching processes. It is precisely in the context ofbranching processes with bounded off-spring that T. Harris, in 1948, first setforth the possibility that the logarithm of the moment generating function ofthe rescaled population size, in the super-critical regime, does not grow nearinfinity as a power, but it has an oscillatory prefactor. These oscillationshave been observed numerically only much later and, while the origin is clearlytied to the discrete character of the iteration, the amplitude size is not sowell understood. The purpose of this note is to reconsider the issue forhierarchical models and in what is arguably the most elementary setting -- thepinning model -- that actually just boils down to iteration of polynomial maps(and, notably, quadratic maps). In this note we show that the oscillatorycritical amplitude for pinning models and the oscillating pre factor connectedto the Harris random variable coincide. Moreover we make explicit the linkbetween these oscillatory functions and the geometry of the Julia set of themap, making thus rigorous and quantitative some ideas set forth by B. Derrida,C. Itzykson and J. M. Luck (CMP 1984).

Oscillatory critical amplitudes in hierarchical models and the tail of the Harris random variable