# Edge fluctuations of limit shapes

## TL;DR

This paper surveys the boundary fluctuations of limit shapes in random tiling and dimer models, highlighting the universal Airy process and the determinantal structure enabling detailed analysis.

## Contribution

It provides a comprehensive overview of models with determinantal point processes and discusses the types of limit laws that can be derived from their correlation kernels.

## Key findings

- Boundary fluctuations follow universal limit laws like the Airy process.
- Determinantal point processes facilitate detailed analysis of limit shapes.
- Various models exhibit similar fluctuation behaviors due to underlying mathematical structures.

## Abstract

In random tiling and dimer models we can get various limit shapes which gives the boundaries between different types of phases. The shape fluctuations at these boundaries give rise to universal limit laws, in particular the Airy process. We survey some models which can be analyzed in detail based on the fact that they are determinantal point processes with correlation kernels that can be computed. We also discuss which type of limit laws that can be obtained.