Before revision:  In STOC'18, Assadi et al. presented a distributed algorithm for maintaining an MIS with O(1) ( amortized) round complexity and O(m^{3/4\left(\log^2 n}\right) \left(\right)  ]  Assadi et al. is deterministic;
After revision:  In STOC'18, Assadi et al. presented a distributed algorithm for maintaining an MIS with O(1) ( amortized) round complexity and O(m^{3/4\left(\log^2 n}\right) \left(\right) ] up to a polylog n factor. The previous best deterministic distributed MIS algorithm, by Assadi et al. is deterministic;