SciPost Submission Page

Extreme boundary conditions and random tilings

by Jean-Marie Stéphan

Submission summary

As Contributors:	Jean-Marie Stéphan
Arxiv Link:	https://arxiv.org/abs/2003.06339v1 (pdf)
Date submitted:	2020-03-28 01:00
Submitted by:	Stéphan, Jean-Marie
Submitted to:	SciPost Physics Lecture Notes
Academic field:	Physics
Specialties:	Mathematical Physics Statistical and Soft Matter Physics
Approach:	Theoretical

Abstract

Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]

---