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Recent progress in multi-type WR models

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The q-type WR (Widom—Rowlinson) model describes q species of particles/cells in d dimensions, repelling each other when they belong to different species (a hard-core repulsion) and tolerating each other when they belong to the same species. For d at least 2 and q=2 (the original form of the model), if the densities of each type are high (and equal), the model is an example of non-uniqueness of shift-invariant ergodic Gibbs/DLR measures (Ruelle, 1972). Namely, there exist two such measures, they are generated by corresponding boundary conditions and are transformed into each other by interchanging the types of particles.

For larger values of q, the picture is more complicated and depends on the collection of hard-core repulsion diameters. In the talk, I’ll give a complete solution for q=3,4 and discuss conjectures for q at least 5. If the time permits, I’ll touch on quantum aspects of the problem.

This is a joint work with A. Mazel and I. Stuhl. No special preliminary knowledge is expected from the audience, either in Probability Theory or in Statistical Mechanics.

This talk is part of the Probability series.

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