Noncommutative geometry of random surfaces

TL;DR

This paper introduces a noncommutative geometric framework for analyzing random surfaces modeled by bipartite planar dimers, providing new insights into their correlation structures and quantization of limit shapes.

Contribution

It constructs a noncommutative curve associated with bipartite dimer models, linking it to inverse Kasteleyn matrices and extending the limit shape theory through quantization.

Findings

Associates a noncommutative curve to bipartite dimer models

Determines all correlations via the inverse Kasteleyn matrix

Proposes a quantization of the limit shape construction

Abstract

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape construction of Kenyon and the author in arXiv:math-ph/0507007. We also discuss various directions in which this correspondence may be generalized.