Michael Aizenman: A new perspective on a pair of two dimensional phenomena

The full title is: A new perspective on a pair of two dimensional phenomena: delocalization in random height functions and the Berezinskii-Kosterlitz-Thouless phase of O(2) symmetric spin models

Delocalization in integer-restricted Gaussian field, and other random height functions formulated over planar doubly-periodic lattices, is shown to imply slow (power law) decay of correlations in the corresponding dual O(2) symmetric two-component spin model. The link proceeds through a lower bound on the spin-spin correlation in terms of the probability of their sites being on a common level loop of the dual random height function. Motivated by this observation, we have extended the recent proof by P. Lammers of delocalization transition in two dimensional graphs of degree three, to all doubly periodic graphs, in particular to Z^2. The extension is established through a monotonicity argument based on lattice inequalities of O. Regev and N. Stephens-Davidowitz. Taken together the results yield a new perspective on the BKT phase transition in O(2)-invariant models and complete a new proof of delocalization in two-dimensional integer-valued height functions. (Both phenomena are unique to two dimension, and have been previously proven and studied by other means). That talk is based on a joint work by M. Aizenman, M. Harel J. Shapiro and R. Peled.

One world IAMP mathematical physics seminar