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### Monotonicity theorems for integer-valued fields and delocalization in two-dimensions

**QMAP Seminar**

Speaker: | Jacob Shapiro, Princeton University |

Location: | 3024 PSEL/QMAP |

Start time: | Fri, May 13 2022, 12:10PM |

Integer-valued fields are restricted to take values in Z and usually their Gibbs factor depends only on the gradient of the field. When the Gibbs factor is such that the typical value of the gradients is much larger than 1 (the spacing of points in Z), the integer constraint becomes less relevant so the field behaves as if it were real-valued and “delocalizes”. In 2D, this delocalization is associated with the Berezinskii–Kosterlitz–Thouless phase of the dual O(2) spin model. I will explain these notions for various models and present recent monotonicity theorems for fluctuations which are important to establish the delocalized phase.

Joint with: Michael Aizenman, Matan Harel and Ron Peled.