## Periodic Reporting for period 4 - PTRCSP (Phase Transitions in Random Constraint Satisfaction Problems)

Reporting period: 2022-10-01 to 2023-09-30

Although the current knowledge about random discrete structures is broad, there are many fundamental and long-standing questions with respect to the key characteristics. In particular, up to a small number of notable exceptions, several well-studied models undoubtedly exhibit phase transitions, but we do not understand them from a mathematical viewpoint. The goal of the proposed project was to study some prominent open problems whose solution will improve significantly our general understanding of phase transitions and of the fine behaviour in random discrete structures. The objectives included the establishment of phase transitions in random constraint satisfaction problems and the analysis of the limiting distribution of central parameters, like the chromatic number.

The project resulted in 17 publications, some of which were published in the highest-ranked journals in the field. One notable highlight is the work on the non-concentration of the chromatic number of random graphs, particularly addressing a longstanding question in random graph theory. The results, including a polynomial non-concentration bound and a two-point concentration bound of specific types of colourings, have significantly advanced the state of the art and have also opened new avenues for future research. Additionally, progress was made in understanding phase transitions in random constraint satisfaction problems, with extensive analyses, determinations of satisfiability thresholds, and significant contributions to decoding problems and information theory, culminating in the rigorous grounding of the condensation phase transition in a large class of models. The project's impact extends to algorithmic advancements and the study of evolving structures, showcasing unexpected hitting time results and furthering our understanding.

Let me start with a result that stands out. As decribed in the proposal, one of the big open questions in random graph theory is to understand the distribution of the chromatic number of random graphs with a quadratic number of edges. In particular, the concentration is an important parameter - what is the size of the smallest interval in which this quantity typically attains values? - and no reasonable lower bounds were known, although the problem is more than 60 years old. Annika Heckel showed within the project a polynomial non-concentration bound, a fantastic result that will make it into the history books and that was published in the Journal of the Americal Mathematical Society.

Actually, we made much more progress with respect to the investigation of the chromatic numner. In a tour de force paper we established one of the main hypotheses that were postulated in the proposal. In particular, we established that if we colour a random graph in a way that each colour is used at most a-2 times, where a is the independence number (that is, the maximum number of times a colour can be used), then the distribution of the chromatic number concentrates in most cases on a single value. Moreover, several intermediate milestones were reached. A particular example is the question about the probability of non-existence in random subsets, which was treated in a very broad and generic setting and the results were published in the Annals of Probability. The research has not reached its end yet and we have discovered new and promising avenues: in a follow up paper we will use all this and many more new ideas to establish the sought limiting distribution for the chromatic number for a wide range of the parameters.

The other research objectives of the project were about the establishment of phase transitions in random constraint satisfaction problems. In this direction we achieved several results. First, the satisfiability threshold for various problems was investigated and determined. In particular, we considered so-called occupation problems on regular graphs; such problems are central and are encountered repeatedly, although mostly in disguise. We studied the property of orientability for different models of random hypergraphs, contagion mechanisms in inhomogeneous random graphs, dispersion processes and cluster statistics/component sizes.

All the previously mentioned problems turn out to belong to the so-called „replica symmetric“ setting in the language of the proposal; the analysis was in all cases challenging and non-trivial optimization problems had to be considered and solved. Regarding the analysis of problems that are not replica symmetric we composed within the project two tour-de-force papers, each one comprising of nearly 100 pages, where we considered a rather general model (in some settings, more general than described in the proposal) of random factor graphs that includes inference problems such as decoding problems or the stochastic block model. The key quantity from an information-theoretic perspective is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created. For example, in the stochastic block model, this is be the partition that we planted initially (the ‚blocks‘). The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs, we verified a formula for the mutual information predicted by physics techniques. As a main application and consequence, we proved an important conjecture about low-density generator matrix codes. We also presented many other applications, including phase transitions of the stochastic block model and the mixed k-spin model from physics. These papers put the condensation phase transition (where the solution space suddenly loses its ‚symmetry‘ and solutions start to gather around specific points) on a firm rigorous ground.

Apart from these results, there are also other results that were achieved within the project. For instance, there were algorithms-related results in the project, and results that address structures that rather than being static, they evolve over time, for example by adding constraints one by one. Several strong and not expected hitting time results were shown in this context.