The critical length for growing a droplet

Journal article

In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of ‘droplets’, and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper we construct a mechanism for the growth of droplets in an arbitrary finite-range monotone cellular automaton on a d-dimensional lattice. Our main application is an upper bound on the critical probability for percolation that is sharp up to a constant factor in the exponent. Our method also provides several crucial tools that we expect to have applications to other interacting particle systems, such as kinetically constrained spin models on Zd .

This is one of three papers that together confirm the Universality Conjecture of Bollob´as, Duminil-Copin, Morris and Smith.

Authors: Balister, P; Bollobás, B; Morris, R; Smith, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: American Mathematical Society
• Journal: Memoirs of the American Mathematical Society
• Volume: 310
• Issue: 1571
• Publication date: 2025-06-10
• DOI: 10.1090/memo/1571
• EISSN: 1947-6221
• ISSN: 0065-9266
• EISBN: 9781470481896
• ISBN: 9781470474874
• Language: English