Oscillatory Critical Amplitudes in Hierarchical Models and the Harris Function of Branching Processes

Abstract

Oscillatory critical amplitudes have been repeatedly observed in hierarchical models and, in the cases that have been taken into consideration, these oscillations are so small to be hardly detectable. Hierarchical models are tightly related to iteration of maps and, in fact, very similar phenomena have been repeatedly reported in many fields of mathematics, like combinatorial evaluations and discrete branching processes. It is precisely in the context of branching processes with bounded off-spring that T. Harris, in 1948, first set forth the possibility that the logarithm of the moment generating function of the rescaled population size, in the super-critical regime, does not grow near infinity as a power, but it has an oscillatory prefactor. These oscillations have been observed numerically only much later and, while the origin is clearly tied to the discrete character of the iteration, the amplitude size is not so well understood. The purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting -the pinning model -that actually just boils down to iteration of polynomial maps (and, notably, quadratic maps). In this note we show that the oscillatory critical amplitude for pinning models and the oscillating pre factor connected to the Harris random variable coincide. Moreover we make explicit the link between these oscillatory functions and the geometry of the Julia set of the map, making thus rigorous and quantitative some ideas set forth in [10].

Log-periodic Critical Amplitudes: A Perturbative Approach

“…It is rather straightforward to extract from this formula that ψ ∞ (·) is (real) analytic on (x * , ∞) (e.g. [8]): in statistical mechanics terms this says that x * is the only critical point (note that with the definition (1.2, 1.3) one has ψ ∞ (x) = 0 for x ∈ (0, x * ]).…”

“…Harris was unable to establish that L(·) is not a constant as soon as q p < 1 (for q p = 1, that is for F (x) = x p , it is straightforward to see that L(·) is constant) and this open issue has drawn the attention in the mathematical community (see for example [5,6], dealing precisely with the problem left open by Harris) and similar -sometimes strictly related -oscillatory behaviors have been pointed out repeatedly (e.g. [27,17,33,22,37] and [8] for further references: the stress is often on the nearly constant and nearly sinusoidal character of these oscillations). To our knowledge, establishing in general that L(·) is non constant is still an open problem for p > 2 (for p = 2 the non triviality of L(·) is established in [8] by exploiting results in [10]).…”

“…[27,17,33,22,37] and [8] for further references: the stress is often on the nearly constant and nearly sinusoidal character of these oscillations). To our knowledge, establishing in general that L(·) is non constant is still an open problem for p > 2 (for p = 2 the non triviality of L(·) is established in [8] by exploiting results in [10]). …”

“…It is straightforward to see that the limit satisfies the functional equation 5) which goes under the name of Böttcher equation in the iterated function literature [3,30], and one expects and sometime can prove -see for example [5,8] -that 6) where the exponent α is usually non-integer and B(·) is a strictly positive periodic function of period one.The exponent α has the well known expression…”

“…(1.6) is easily guessed: if one assumes that ψ ∞ (x) ∼ (x − x * ) α C(x − x * ) as x ց x * , with y → C(y) defined for y ∈ (0, y 0 ) for an arbitrary choice of y 0 > 0, continuous and such that 0 < C min < C(y) < C max , one gets from (1.5) that 8) and this implies the value (1.7) of α and the fact C F ′ (x * )y = C(y) for every y, that is that C(·) is multiplicatively periodic of period F ′ (x * ). Equivalently, C(y) = B(−log(x − x * )/ log F ′ (x * )), with B(·) periodic of period one, as in (1.6).…”