Bratteli diagrams where random orders are imperfect

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 $B$, a random order on $B$ does not admit a continuous Vershik map.