Reversible Poisson-Kirchhoff Systems

TL;DR

This paper introduces a reversible class of weighted line systems on the quarter plane, generalizing classical models like last passage percolation and the six-vertex model, with new invariant measures and reversible properties.

Contribution

It defines a new class of weighted Poisson-Kirchhoff systems that extend existing models, incorporating weights and Markovian dynamics that preserve invariance and reversibility.

Findings

Derived explicit invariant measures for certain models

Established new reversible properties for six-vertex models

Generalized classical processes with weighted line systems

Abstract

We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.