A phase transition in a spatial permutation model on infinite trees

Seminar | February 6 | 3-4 p.m. | 1011 Evans Hall

 Milind Hegde, UC Berkeley

 Department of Statistics

Abstract: Spatial random permutation models are of physical interest due to connections to representations of certain gases such as helium as well as of the quantum Heisenberg ferromagnet. Physical phase transitions in these contexts correspond to the appearance of macro or infinite cycles in the permutation model. We study a spatial random permutation model on infinite trees with a time parameter T, a special case of which is the random stirring or random interchange process. The model on trees was first analysed by Björnberg and Ueltschi, who established the existence of infinite cycles for T slightly above a putatively identified critical value but left open behaviour at arbitrarily high values of T. We show the existence of infinite cycles for all T greater than a constant, thus classifying behaviour for all values of T and establishing the existence of a sharp phase transition. Our argument analyses a variant of simple random walk on the tree which is closely related.

Work with Alan Hammond