  The free energy of the Random Energy Model at the transition point between
ferromagnetic and spin glass phases is calculated. At this point, equivalent to
the decoding error threshold in optimal codes, free energy has finite size
corrections proportional to the square root of the number of degrees. The
response of the magnetization to the ferromagnetic couplings is maximal at the
values of magnetization equal to half. We give several criteria of complexity
and define different universality classes. According to our classification, at
the lowest class of complexity are random graph, Markov Models and Hidden
Markov Models. At the next level is Sherrington-Kirkpatrick spin glass,
connected with neuron-network models. On a higher level are critical theories,
spin glass phase of Random Energy Model, percolation, self organized
criticality (SOC). The top level class involves HOT design, error threshold in
optimal coding, language, and, maybe, financial market. Alive systems are also
related with the last class. A concept of anti-resonance is suggested for the
complex systems.
