Concentration and threshold phenomena in random graphs and hypergraphs

Informations projet

RandomHypGra

N° de convention de subvention: 101044123

DOI

10.3030/101044123

Date de signature de la CE 7 Avril 2022

Date de début 1 Octobre 2022

Date de fin 30 Septembre 2027

Financé au titre de

• European Research Council (ERC)

Coût total

€ 1 621 875,00

Contribution de l’UE

€ 1 621 875,00

Coordonné par

TEL AVIV UNIVERSITY
Israel

Periodic Reporting for period 1 - RandomHypGra (Concentration and threshold phenomena in random graphs and hypergraphs)

Période du rapport: 2022-10-01 au 2025-03-31

Résumé du contexte et des objectifs généraux du projet

The central theme of this research project is the emergence of 'global' structure in large random objects that are defined using a family of 'local' constraints. There are numerous well-studied instances of this abstract problem in mathematics, theoretical computer science, and mathematical physics. This research project aims to study a number of interrelated open problems spanning several branches of pure mathematics (combinatorics, Ramsey theory, and probability theory) that focus on different aspects of one unifying theme. The four main avenues of research are:

(i) The study of the global structure of a random graph conditioned on the unlikely event that the number of copies of some fixed subgraph exceeds its expectation.
(ii) The search for a general-purpose method that generalises the so-called hypergraph container lemma.
(iii) The search for an extension of the cluster expansion method in statistical physics to allow for k-wise interactions (k > 2).
(iv) Threshold phenomena associated with Turán-type and Ramsey-type problems.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

We have made substantial progress towards a range of goals set out in the proposal. The following are the most notable developments/achievements:

(i) Resolution of the Kohayakawa–Kreuter conjecture from 1997, which predicts the location of the threshold for the emergence of the asymmetric Ramsey property in the binomial random graph G(n,p) for an arbitrary sequence of graphs (H_1, ... H_r).
(ii) Determination of the sharp threshold for the exact analogue of Turán's theorem in G(n,p).
(iii) Complete description of the upper-tail deviations for the number of k-term arithmetic progressions in random subsets of integers.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

The two most significant results that go beyond the state of the art are:
(i) The development of a novel method of analysing largest and nearly-largest cuts in random graphs.
(ii) Two new proofs of the hypergraph container lemma that show unexpected connections between the container method, Janson's inequality, the Kahn–Kalai conjecture, and the hard-core model.