Semifinite harmonic functions on the Gnedin–Kingman graph

In 1978 J.F.C. Kingman described random exchangeable partitions of the set of natural numbers. Kingman's result can be reformulated in terms of harmonic functions on some branching graph, which is called the Kingman graph. Vertices of this graph correspond to Young diagrams and edges correspond to the Pieri rule for the monomial basis in the algebra of symmetric functions.
In 1997 A. Gnedin discovered an analog of the Kingman's result for linearly ordered partitions. Gnedin's theorem can be reformulated in terms of harmonic functions on some branching graph too. We will call this graph the Gnedin-Kingman graph. Its vertices correspond to compositions (ordered partitions) and edges correspond to Pieri rule for the monomial basis in the algebra of quasisymmetric functions.
The talk is devoted to indecomposable seminfinite harmonic functions on the Gnedin-Kingman graph. The semifinitness property means that the value of a function on some vertices equals $+\infty$. We will also discuss multiplicativity of indecomposable semifinite harmonic functions on the Gnedin-Kingman graph and how they link to semifinite harmonic functions on the Kingman graph. The latter were classified by A. Vershik and S. Kerov in 1980's.