Wednesday, April 17th, 2013
2:00pm - 3:00pm, in McCormack 2-116

Mirjana Vuletic

UMass Boston

The Gaussian free field and strict plane partitions

Abstract: We study asymptotic properties of a random model on combinatorial structures called strict plane partitions. The model can also be seen as a random surface model, random domino tiling model or point process (a measure on finite subsets of an integer lattice). The model has a limit shape, which, very loosely, means that there is the expected shape of large strict plane partitions distributed according to the given measure. In this talk the emphasis will be on the height fluctuations around the limit shape, which we have shown are governed by the Gaussian free field. The Gaussian free field is an import class of Gaussian processes that has been associated with random surface and random matrix models, as well as quantum field theory and Schramm-Loewner evolutions. The asymptotic analysis of the model is based on the correlation function, which can be written as a Pfaffian of a matrix whose elements are given by a function called correlation kernel; such random models are called Pfaffian point processes. The results are then derived from the Pfaffian structure and steepest descent analysis of the correlation kernel. In the talk, I will explain how the Gaussian free field arises in our model. The argument can be generalized to a class of Pfaffian processes whose kernels possess certain properties.