Models of percolation based on random walks
Final Report Abstract
1. It has been proved that the random walk loop soup in dimensions d ≥ 3 satisfies a useful decoupling inequality. It has been shown that in a class of strongly correlated percolation models that satisfy such a decoupling inequality, the unique infinite cluster has properties similar to those of uncorrelated percolation; for example, the random walk on the infinite cluster satisfies the quenched invariance principle and the quenched Gaussian heat-kernel bounds. Both the random walk loop soup and its vacant set are in this class. 2. It has been proved that the Poisson cylinder’s percolation in dimensions d ≥ 3 satisfies a useful decoupling inequality; consequently, it has been shown that the occupied set is almost surely transient. 3. The sharpness of percolation phase transition has been shown for some planar dynamical models of percolation, including a class of opinion dynamics models and the Glauber dynamics for the Ising model.
Publications
- Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup, Electron. J. Probab. 24 (2019), Paper No. 110, 34 pp.
Caio Alves and Artem Sapozhnikov
(See online at https://doi.org/10.1214/19-EJP360) - Percolation phase transition on planar spin systems
Caio Alves, Gideon Amir, Rangel Baldasso, Augusto Teixeira
(See online at https://doi.org/10.48550/arXiv.2105.13314) - Sharp threshold for two-dimensional majority dynamics percolation. Ann. Inst. H. Poincaré Probab. Statist. 58 (4), 1869-1886, (November 2022)
Caio Alves and Rangel Baldasso
(See online at https://doi.org/10.1214/21-AIHP1232)
