Final Report Summary - PTCC (Phase transitions and computational complexity) Randomly generated discrete structures such as random graphs, formulas or codes have played an important role in mathematics for the past half-century. Moreover, since the 1990s it has been claimed based on experimental evidence that some of these objects pose extremely challenging algorithmic benchmarks for fundamental problems in computer science. Yet many of the properties of random discrete structures remained shrouded in mystery. This concerns particularly the location and nature of "phase transitions". These are parameter values at which the properties of a discrete structure undergo a sudden qualitative change.Over the past 20 years physicists have developed a sophisticated technique for the study of random discrete structures called the "cavity method". This technique leads to remarkable predictions about phase transitions in discrete structures and their impact on computational problems. In addition, the cavity method has been used to put forward new "message passing algorithms" that perform very well experimentally. But the physics approach does not satisfy the standard of mathematical rigor that is commonly required in mathematics and in the theory of computing. Hence, the aim of this project is to put the predictions of the cavity method on a rigorous footing and to explore the impact on mathematics and computer science with rigorous methods.The project achieved several main results. We developed new rigorous methods for the study of random discrete structures, their phase transitions and the probability distributions that they induce. This enabled us to verify several important predictions of the cavity method such as the existence of a so-called "condensation phase transition" in the random graph colouring problem, one of the main benchmark problems in the theory of random discrete structures. Additionally, we obtained results on the information-theoretic thresholds in inference problems. Further, we established a connection between a decorrelation property and the Belief Propagation message passing scheme and furnished a rigorous analysis of a basic version of the Survey Propagation message passing algorithm for the random k-SAT problem.
