
Research Grant ``Supercritical behaviour in random subgraph models''
Summary
Percolation, or random subgraphs, is a mathematical model originally studied in the context of statistical physics, where they model the flow of a liquid or gas through a lattice like medium whose channels are randomly blocked. For many of these models, as the density of the random subgraph increases, there is a threshold at which its likely structure changes dramatically. Below this threshold all the components are small, whereas above this threshold many of these small component coalesce and a unique large component appears.
In this supercritical regime, whilst the random subgraph is still quite sparse and disconnected, its largest component displays many interesting structural properties which you would expect to appear only for much denser graphs. This project aims to investigate the structural properties of these supercritical random subgraphs, and in particular their largest components, in a range of percolation models.
Grant Info
Supported by Austrian Science Fund (FWF), Grant no. P36131, 01.01.202331.12.2025
Team
 Mauricio Collares
 Joshua Erde (PI)
Collaborators and visitors
 Sahar Diskin, Tel Aviv University
 Michael Krievelevich, Tel Aviv University
 Matthew Jensen, King's College London
 Mareclo Campos, Oxford University
Publications
Articles in Journals

(Lucas Aragão, Maurício Collares, João Pedro Marciano, Taísa Martins and Robert Morris) A lower bound for setcoloring Ramsey numbers, Random Structures and Algorithms, Volume 64, Issue 2, 2023 (Journal/arXiv).
(N. Bowler, C. Elbracht, J. Erde, J. P. Gollin, K. Heuer, M. Pitz and M. Teegen) Ubiquity of graphs with nonlinear end structure, Journal of Graph Theory, Volume 103, Issue 3, 2023 (Journal/arXiv).

(T. Do, J. Erde, M. Kang and M. Missethan) Component behaviour and excess of random bipartite graphs near the critical point, Electronic Journal of Combinatorics, Volume 30, Issue 3, 2023 (Journal/arXiv).
(S. Diskin, J. Erde, M. Kang and M. Krivelevich) Percolation on irregular highdimensional product graphs, Combinatorics, Probability and Computing, 2023 (Journal/arXiv).

(B. Barber, J. Erde, P. Keevash and A. Roberts) Isoperimetric stability in lattices, Proceedings of the American Mathematical Society, Volume 151, 2023 (Journal/arXiv).

(Marcelo Campos, Maurício Collares, Guilherme Oliveira Mota) Counting orientations of random graphs with no directed kcycles, Random Structures and Algorithms, to appear (Journal/arXiv).
(S. Diskin, J. Erde, M. Kang and M. Krivelevich) Isoperimetric inequalities and supercritical percolation on highdimensional product graphs, Combinatorica, to appear (arXiv).

(T. Do, J. Erde and M. Kang) A note on the width of sparse random graphs, Journal of Graph Theory, to appear (Journal/arXiv).
(S. Diskin, J. Erde, M. Kang and M. Krivelevich) Isoperimetric inequalities and supercritical percolation on highdimensional product graphs, Combinatorica, to appear (arXiv).
(J. Erde, F. Lehner, M. Kang, D. Schmid and B. Mohar) Catching a robber on a random kuniform hypergraph, Canadian Journal of Mathematics, to appear (arXiv).
Articles in PeerReviewed Conference Proceedings

(Lucas Aragão, Maurício Collares, João Pedro Marciano, Taísa Martins and Robert Morris) A lower bound for setcoloring Ramsey numbers, Extended Abstracts EuroComb 2023 2023 (Proceedings).

(J. Erde, F. Lehner, M. Kang, D. Schmid and B. Mohar) Cop number of random kuniform hypergraphs, Extended Abstracts EuroComb 2023 2023 (Proceedings).

(B. Barber, J. Erde, P. Keevash and A. Roberts) Isoperimetric stability in lattices, Extended Abstracts EuroComb 2023 2023 (Proceedings).
last updated in January 2023
