Limit Theory in Combinatorics

Informations projet

CombLimit

N° de convention de subvention: 750857

DOI

10.3030/750857

Projet clôturé

Date de signature de la CE 24 Février 2017

Date de début 1 Mars 2017

Date de fin 28 Février 2019

Financé au titre de

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coût total

€ 134 239,20

Contribution de l’UE

€ 134 239,20

Coordonné par

HUN-REN RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Hungary

Periodic Reporting for period 1 - CombLimit (Limit Theory in Combinatorics)

Période du rapport: 2017-03-01 au 2019-02-28

Résumé du contexte et des objectifs généraux du projet

Large networks are present everywhere is our world. We have communication networks, traffic networks, social networks, we use our network of brain cells, etc.
Graphs are the mathematical abstraction of networks. Properties of large graphs are analyzed by their limit when the size tends to infinity.
This is analogous to how a physical medium is defined as a continuous object describing the case when the number of particles tends to infinity.
This is a very recent area of mathematics and the goal is to contribute to its general mathematical theory.
This will lead to a better understanding of our world.

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 extended our understanding of random graphs to locally transitive graphs via inequalities of IID-processes.
The problem of finding an almost largest independent set in random graphs is a benchmark problem of the topic, and we made a significant contribution about the problem.
For example, while we know that almost maximum size independent sets cannot be constructed locally in random regular graphs of large degree, now we get closer to prove that it is not true for small degree.
This leads to the understanding of an unexpected phase transition, a change in the structure of random graphs.

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 project initiated a new research direction. Statistical physicists have non-rigorous (and sometimes wrong) conjectures about the structures of random graphs.
In the recent years, some of these intuitions are transformed to mathematical proofs by extremely long mathematical papers.
But we realized that some of these ideas are similar to ours, and it would be very beneficial to connect these areas.
A new research group about these problems has already formed at Rényi Institute. Our first big but very plausible goal is to understand, simplify and generalize the 78 pages long recent Acta Mathematica paper about the independence ratio of random regular graphs. The ultimate goal is to connect graph limit theory with statistical physics, hereby discovering and deeply understanding unknown phase transitions, and hereby getting a better understanding the structures of large graphs. This may give new explanation of real-life phenomena such as ``regime changes'' or aging.

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