Fermionic Wick rule in planar statistical models

Abstract

In this thesis we analyze the implications of planarity on the structure of the correlation functions of the Ising model and the dimer model, using geometrical techniques. In the first chapter, we discuss the random current representation on the Ising model, and how it can be used to extract information about the 2𝑛-point correlation functions. We show the proof of fermionic structure for boundary spins. The interest of this technique is that it doesn’t depend on the specific details of the model, but on the topological implications of planarity. In the second chapter, we inspect the dimer model using a similar representation of the correlation functions in terms of paths and loops, obtained by superimposing two different matchings (double dimer model). This leads us to an analogous result for boundary correlators. In the last chapter, we look for an extension of this results for bulk spins. In some recent developments [3], it has been shown that orderdisorder operators, which can be seen as a generalization of boundary spins, obey the Wick rule in the Ising model. We present a suggestion on how to introduce this operators on the dimer model. Our main result is the proof of the fermionic structure of the correlators of such variables on planar graphs.