Random skew plane partitions with a piecewise periodic back wall

TL;DR

This paper studies the asymptotic behavior of large random skew plane partitions with a piecewise linear boundary, revealing new smooth limit shape boundaries and local fluctuation phenomena.

Contribution

It extends previous work by analyzing non-lattice slopes, showing the limit shape boundary is smooth, and identifying the bead process in local fluctuations.

Findings

Limit shape boundary is a smooth curve.

Identification of the bead process in local fluctuations.

New behavior in the asymptotics compared to previous models.

Abstract

Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in [OR2], but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in [OR2] is singular. We also observe the bead process introduced in [B] appearing in the asymptotics at the top of the limit shape.