Bratteli diagrams where random orders are imperfect

TL;DR

This paper investigates the properties of random orders on simple Bratteli diagrams, revealing a growth rate threshold that determines the existence of infinite paths and continuous Vershik maps.

Contribution

It establishes a dichotomy linking growth rates of Bratteli diagrams to the structure of random orders and Vershik maps.

Findings

Uncountably many infinite paths occur with super-linear growth.

Random orders on slowly growing diagrams admit homeomorphisms.

Most random orders on rapidly growing diagrams lack continuous Vershik maps.

Abstract

For the simple Bratteli diagrams B where there is a single edge connecting any two vertices in consecutive levels, we show that a random order has uncountably many infinite paths if and only if the growth rate of the level-n vertex sets is super-linear. This gives us the dichotomy: a random order on a slowly growing Bratteli diagram admits a homeomorphism, while a random order on a quickly growing Bratteli diagram does not. We also show that for a large family of infinite rank Bratteli diagrams, a random order on B does not admit a continuous Vershik map.