Research on the formation and fluctuation of random shapes in mathematical models of statistical mechanics

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1. We found a new proof of the fact that there are only two extremal Gibbs states for the two dimensional Ising model based on the percolation argument. As an application of this new method, we proved that there are only two extremal points of translationally invariant Gibbs states for the two dimensional Widom-Rowlinson model for sufficiently low temperatures.2. We gave an estimate of the speed of convergence for the time constant of the first passage Ising percolation for temperatures above the critical point.3. We proved a Dobrushin-Hryniv type limit theorem for the two-dimensional Widom-Rowlinson model. The conditions for the result to hold are a little relaxed.