Information, Physics, and Computation
This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. It is accessible to graduate students and researchers without a specific training in any of these fields. The selected topics include spin glasses, error correcting codes, satisfiability, and are central to each field. The approach focuses on large random instances and adopts a common probabilistic formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to study phase transitions.
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1RSB cavity algorithm assume asymptotic average behaviour belief propagation Bethe measures binary Boltzmann distribution bound BP equations cavity method channel Chapter clause clusters codeword compute condition configuration Consider converges correlation corresponding cost defined denote density evolution dynamics edges energy function entropy density estimate Exercise exists expectation exponentially factor graph ferromagnetic finite fixed point formula free energy free entropy free-entropy density function nodes given graphical model Hamming distance high probability instance integer Ising model iterations limit log2 magnetic MAP decoding marginals Markov chain matrix messages min-sum obtained optimal partition function phase transition polynomial probability distribution problem random graph random variables replica replica-symmetric result saddle point sample SAT assignments satisfiability Section Show solution spin glass subset symmetric temperature Theorem thermodynamic threshold tree variable nodes vertex vertices XORSAT