T-embeddings as stationary measures on T-graphs

In this paper, we investigate the relation between t-embeddings and T-graphs associated with the same weighted planar bipartite graph. We prove that, given such a T-graph, the corresponding t-embeddings are classified by positive stationary measures on this T-graph and describe a reconstruction procedure of the t-embedding given such a measure. As an application, we develop an approach for constructing t-embeddings of arbitrary sufficiently non-degenerate planar bipartite graphs.