Alcove path model for $B(\mathrm{\infty })$

We construct a model for $B(\mathrm{\infty })$ using the alcove path model of Lenart and Postnikov. We show that the continuous limit of our model recovers a dual version of the Littelmann path model for $B(\mathrm{\infty })$ given by Li and Zhang. Furthermore, we consider the dual version of the alcove path model and obtain analogous results for the dual model, where the continuous limit gives the Li and Zhang model.