## Periodic Reporting for period 3 - PTRCSP (Phase Transitions in Random Constraint Satisfaction Problems)

Reporting period: 2021-04-01 to 2022-09-30

The systematic investigation of random discrete structures and processes was initiated by Erdős and Rényi in a seminal paper about random graphs in 1960. Since then the study of such objects has become an important topic that has remarkable applications not only in combinatorics, but also in computer science and statistical physics.

Random discrete objects have two striking characteristics. First, they often exhibit phase transitions, meaning that only small changes in some typically local control parameter result in dramatic changes of the global structure. Second, several statistics of the models concentrate, that is, although the support of the underlying distribution is large, the random variables usually take values in a small set only. A central topic is the investigation of the fine behaviour, namely the determination of the limiting distribution.

Although the current knowledge about random discrete structures is broad, there are many fundamental and long-standing questions with respect to the two key characteristics. In particular, up to a small number of notable exceptions, several well-studied models undoubtedly exhibit phase transitions, but we are not able to understand them from a mathematical viewpoint nor to investigate their fine properties. The goal of the proposed project is 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 include the establishment of phase transitions in random constraint satisfaction problems and the analysis of the limiting distribution of central parameters, like the chromatic number in dense random graphs.

Random discrete objects have two striking characteristics. First, they often exhibit phase transitions, meaning that only small changes in some typically local control parameter result in dramatic changes of the global structure. Second, several statistics of the models concentrate, that is, although the support of the underlying distribution is large, the random variables usually take values in a small set only. A central topic is the investigation of the fine behaviour, namely the determination of the limiting distribution.

Although the current knowledge about random discrete structures is broad, there are many fundamental and long-standing questions with respect to the two key characteristics. In particular, up to a small number of notable exceptions, several well-studied models undoubtedly exhibit phase transitions, but we are not able to understand them from a mathematical viewpoint nor to investigate their fine properties. The goal of the proposed project is 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 include the establishment of phase transitions in random constraint satisfaction problems and the analysis of the limiting distribution of central parameters, like the chromatic number in dense random graphs.

In a team consisting of the principal investigator, two postdocs and one PhD student we worked intensively on reaching the objectives of the project. Substantial progress was made in several directions.

A special highlight is a result about the chromatic number of random graphs. One of the big open questions in random graph theory is to understand the distribution of the chromatic number. In particular, the concentration is an important parameter (what is the size of the smallest intervall in which this quantity typically attains values?) and no lower bounds were known, although the problem is more than 60 years old. Annika Heckel showed a poylnomial non-concentration bound, a fantastic result that will make it into the history books.

Other than that, several other steps towards reaching the project goal of determining the limiting distribution of the chromatic number were performed successfully. For example, the probability of non-existence in random subsets is now understood in great detail, and the one-point concentration of specific colorings is established.

Regarding phase transistions in constraint satisfaction problems we made progress towards various goals. The satisfiability threshold for various problems, for example occupation problems and contagion processes, was investigated, in binomial as well as uniform models. All these problems turn out to belong to the so-called „replica symmetric“ setting in the language of the proposal; regarding the analysis of problems that are not replica symmetric we verified in a tour-de-force paper for a very general model of random factor graphs a formula for the mutual information predicted by physics techniques. As a main application and consequence we resolved an important and prominent conjecture about low-density generator matrix codes. Last but not least, we considered algorithms-related questions and evaluated the performance of message-passing algorithms (Belief propagation).

A special highlight is a result about the chromatic number of random graphs. One of the big open questions in random graph theory is to understand the distribution of the chromatic number. In particular, the concentration is an important parameter (what is the size of the smallest intervall in which this quantity typically attains values?) and no lower bounds were known, although the problem is more than 60 years old. Annika Heckel showed a poylnomial non-concentration bound, a fantastic result that will make it into the history books.

Other than that, several other steps towards reaching the project goal of determining the limiting distribution of the chromatic number were performed successfully. For example, the probability of non-existence in random subsets is now understood in great detail, and the one-point concentration of specific colorings is established.

Regarding phase transistions in constraint satisfaction problems we made progress towards various goals. The satisfiability threshold for various problems, for example occupation problems and contagion processes, was investigated, in binomial as well as uniform models. All these problems turn out to belong to the so-called „replica symmetric“ setting in the language of the proposal; regarding the analysis of problems that are not replica symmetric we verified in a tour-de-force paper for a very general model of random factor graphs a formula for the mutual information predicted by physics techniques. As a main application and consequence we resolved an important and prominent conjecture about low-density generator matrix codes. Last but not least, we considered algorithms-related questions and evaluated the performance of message-passing algorithms (Belief propagation).

Although the current knowledge about random discrete structures is very broad, there are many fundamental and long-standing questions. We expect that by the end of this project we will have a significantly better rigorous mathematical understanding of phase transitions in random constraint satisfaction problems and of the limiting distribution of various parameters. Building upon the results that were achieved by now we will determine - at least for a wide range of the parameters - the distribution of the chromatic number of random graphs. Moreover, since we have by now achieved an excellent understanding of the condensation phase transition in constraint satisfaction problems, we will perform the second step and extend the results to the primary goal, namely the satisfiability transition.