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
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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
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Percolation phase transition on planar spin systems
Caio Alves, Gideon Amir, Rangel Baldasso, Augusto Teixeira
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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
