In the Duarte model each vertex of Z <Superscript> 2 can be either infected or healthy. In the bootstrap percolation model version, infected vertices stay infected while a healthy vertex becomes infected if at least two of its North, South, and West neighbors are infected. In the model version with kinetic constraints (KCM), each vertex with rate one changes its state by tossing a biased coin iff the same constraint is satisfied. 

For both versions of the model, an important problem is to determine the divergence as q->0 of the infection time of the origin when the initial infection set is q-random. 

For the bootstrap percolation version, the problem was solved in 2016 by B. Bollobas, H. Duminil-Copin, R. Morris, and P. Smith. For the KCM version, our recent work proves that hidden logarithmically growing energy barriers produce a much sharper divergence. The result also confirms for the Duarte model a universality conjecture for general critical KCM on Z <Superscript> 2 put forward by R. Morris, C. Toninelli and myself.

 Joint work with R. Morris and C. Toninelli (upper bound) and L. Mareche' and C. Toninelli (lower bound).