Homomorphisms from the torus

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• Matthew Jenssen (University of Birmingham)
• Thursday 23 January 2020, 15:00-16:00
• Watson LTA.

If you have a question about this talk, please contact Richard Montgomery.

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus Zⁿ_m, where m is even, to any fixed graph. We show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some “dominant phase”. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper q-colourings of Zⁿ_m (so in particular, the discrete hypercube). For the proof we develop a `Cluster Expansion Method’, which we expect to have further applications, by combining machinery from statistical physics, entropy and graph containers. This is joint work with Peter Keevash.

This talk is part of the Combinatorics and Probability seminar series.

This talk is included in these lists:

• Combinatorics and Probability seminar
• School of Mathematics events
• Watson LTA

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