2001/06/30 by S. Gluzman, D. Sornette, Didier Sornette · 2 citations
Economics, Econometrics and Finance · Physics and Astronomy · #Complex Network Analysis Techniques #Complex Systems and Time Series Analysis #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.65.036142
published as art. no. 036142, Phys. Rev. E V6503 N3 PT2A:U418-U436 (2002) · Latex document of 29 pages + 20 ps figures, addition of a new demonstration of the source of strong log-periodicity and of a justification of the general offered classification, update of reference list
arxiv created 2001/10/04 · openalex publication_date 2002/03/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Log-periodic oscillations have been found to decorate the usual power-law behavior found to describe the approach to a critical point, when the continuous scale-invariance symmetry is partially broken into a discrete-scale invariance symmetry. For Ising or Potts spins with ferromagnetic interactions on hierarchical systems, the relative magnitude of the log-periodic corrections are usually very small, of order 10(-5). In growth processes [diffusion limited aggregation (DLA)], rupture, earthquake, and financial crashes, log-periodic oscillations with amplitudes of the order of 10% have been reported. We suggest a "technical" explanation for this 4 order-of-magnitude difference based on the property of the "regular function" g(x) embodying the effect of the microscopic degrees of freedom summed over in a renormalization group (RG) approach F(x)=g(x)+mu(-1)F(gamma x) of an observable F as a function of a control parameter x. For systems for which the RG equation has not been derived, the previous equation can be understood as a Jackson q integral, which is the natural tool for describing discrete-scale invariance. We classify the "Weierstrass-type" solutions of the RG into two classes characterized by the amplitudes A(n) of the power-law series expansion. These two classes are separated by a novel "critical" point. Growth processes (DLA), rupture, earthquake, and financial crashes thus seem to be characterized by oscillatory or bounded regular microscopic functions that lead to a slow power-law decay of A(n), giving strong log-periodic amplitudes. If in addition, the phases of A(n) are ergodic and mixing, the observable presents self-affine nondifferentiable properties. In contrast, the regular function of statistical physics models with "ferromagnetic"-type interactions at equilibrium involves unbound logarithms of polynomials of the control variable that lead to a fast exponential decay of A(n) giving weak log-periodic amplitudes and smoothed observables.