Friday, May 3, 2013

1305.0139 (Nathanael Berestycki et al.)

Condensation of a two-dimensional random walk and the Wulff crystal    [PDF]

Nathanael Berestycki, Ariel Yadin
We introduce a Gibbs measure on nearest-neighbour paths of length t in the Euclidean planar lattice, where each path is penalized by a factor proportional to the size of its boundary and an inverse temperature \beta. This measure gives a random walk description of the Wulff crystal, representing the distribution of a diluted polymer in a poor solvent. We prove that in two dimensions, if the inverse temperature \beta is larger than the logarithm of the connective constant, then the random walk condensates to a set of diameter t^{1/3}, up to logarithmic corrections. We further speculate that the limiting shape shares the same exponents as the KPZ universality class. A similar result holds for a random walk conditioned to have local time greater than \beta everywhere in its range.
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