Error threshold in optimal coding, numerical criteria, and classes of universalities for complexity.
The free energy of the random energy model at the transition point between the ferromagnetic and spin glass phases is calculated. At this point, equivalent to the decoding error threshold in optimal codes, the free energy has finite size corrections proportional to the square root of the number of degrees. The response of the magnetization to an external ferromagnetic phase is maximal at values of magnetization equal to one-half. We give several criteria of complexity and define different universality classes. According to our classification, at the lowest class of complexity are random graphs, Markov models, and hidden Markov models. At the next level is the Sherrington-Kirkpatrick spin glass, connected to neuron-network models. On a higher level are critical theories, the spin glass phase of the random energy model, percolation, and self-organized criticality. The top level class involves highly optimized tolerance design, error thresholds in optimal coding, language, and, maybe, financial markets. Living systems are also related to the last class. The concept of antiresonance is suggested for complex systems.
Saakian
David B
DB
Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan. saakian@jerewan1.yerphi.am