Condensation of a self-attracting random walk

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• Nathanael Berestycki (Cambridge)
• Tuesday 19 January 2016, 15:00-16:00
• MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.

If you have a question about this talk, please contact Perla Sousi.

I will introduce a Gibbs distribution on nearest-neighbour paths of length t in the Euclidean d-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature \beta. This model can be thought of as a random walk version of the Wulff crystal problem in percolation or the Ising model.

In joint work with Ariel Yadin we prove that, for all \beta>0, the random walk condensates to a set of diameter (t/\beta) ^{{1/3} in dimension d=2, up to a multiplicative constant. In all dimensions d\ge 3, we also prove that the volume is bounded above by (t/\beta)} {d/(d+1)} and the diameter is bounded below by (t/\beta) ^ {1/(d+1)}.

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