Point Fields of Last Passage Percolation and Coalescing Fractional Brownian Motions

TL;DR

This paper investigates point fields arising in KPZ phenomena and coalescing fractional Brownian motions, providing numerical evidence and conjecturing their equivalence in the large-time limit, suggesting a shared universality class.

Contribution

It introduces point fields for coalescing fractional Brownian motions and compares their properties to those in KPZ, proposing a potential universality class connection.

Findings

Numerical evidence shows similar statistical properties between the two point fields.

Conjecture that the point fields are identical in the large-time limit.

Discussion of theoretical support for the universality class hypothesis.

Abstract

We consider large-scale point fields which naturally appear in the context of the Kardar-Parisi-Zhang (KPZ) phenomenon. Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We introduce similarly defined point fields for processes of coalescing fractional Brownian motions (cfBm). The case of the Hurst index 2/3 is of particular interest for us since, in this case, the power law of the density decay is the same as that in the KPZ phenomenon. In this paper, we present strong numerical evidence that statistical properties of points fields in these two different settings are very similar. We also discuss theoretical arguments in support of the conjecture that they are exactly the same in the large-time limit. This would indicate that two objects may, in fact, belong to the same universality class.