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Explore a mathematical lecture examining phase transitions in composition schemes and their applications across combinatorics, number theory, statistical mechanics, and probability theory. Discover how composition schemes exhibit universal behavior in q-enumeration models, where objects with parameter value k have Gibbs measure or Boltzmann weight q^k. Learn about the proof of phase transitions for parameters following Gibbs measures, including the critical value q=q_c where the limit law follows a two-parameter Mittag-Leffler distribution, while Gaussian behavior emerges in the supercritical regime (q>q_c) and Boltzmann distribution appears in the subcritical regime (0
