# Turning point processes in plane partitions with periodic weights of arbitrary period

## TL;DR

This paper analyzes random plane partitions with periodic weights, revealing multiple turning points, facets with rational slopes, and phase transitions, with detailed correlation functions described by GUE-corners processes.

## Contribution

It introduces a model with arbitrary periodic weights, characterizes the boundary phenomena, and computes correlation functions near turning points and phase transitions.

## Key findings

- Multiple turning points correspond to the period of weights.
- Facets with rational slopes form between turning points.
- Correlation functions near turning points relate to GUE-corners processes.

## Abstract

We study random plane partitions with respect to volume measures with periodic weights of arbitrarily high period. We show that near the vertical boundary the system develops up to as many turning points as the period of the weights, and that these turning points are separated by vertical facets which can have arbitrary rational slope. In the lozenge tiling formulation of the model the facets consist of only two types of lozenges arranged in arbitrary periodic deterministic patterns. We compute the correlation functions near turning points and show that the point processes at the turning points can be described as several GUE-corners processes which are non-trivially correlated. The weights we study introduce a first order phase transition in the system. We compute the limiting correlation functions near this phase transition and obtain a process which is translation invariant in the vertical direction but not the horizontal.