Geodesic
The boundary of the refined Kingman graph
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 58-74 Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce the refined Kingman graph $\mathbb D$ whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. We show that the Martin boundary of $\mathbb D$ can be identified with the space $\Omega$ of all sets of disjoint open subintervals of $[0,1]$ and coincides with the minimal boundary of $\mathbb D$. 
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M. V. Karev; P. P. Nikitin. The boundary of the refined Kingman graph. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 58-74. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a5/

[1] J. L. Doob, “Discrete potential theory and boundaries”, J. Math. Mech., 8 (1959), 433–458 | MR | Zbl

[2] A. V. Gnedin, “The representation of composition structures”, Ann. Probab., 25:3 (1997), 1437–1450 | DOI | MR | Zbl

[3] S. V. Kerov, “Combinatorial examples in the theory of AF-algebras”, Zap. Nauchn. Semin. LOMI, 172, 1989, 55–67 | MR | Zbl

[4] S. Kerov, A. Okounkov, G. Olshanski, “The boundary of the Young graph with Jack edge multiplicities”, Int. Math. Res. Not., 4 (1998), 173–199 | DOI | MR | Zbl

[5] A. M. Vershik, “Equipped graded graphs, projective limits of simplices, and their boundaries”, Zap. Nauchn. Semin. POMI, 432, 2015, 83–104 | MR | Zbl

[6] A. M. Vershik, “The problem of describing central measures on the path spaces of graded graphs”, Funct. Anal. Appl., 48:4 (2014), 256–271 | DOI | MR | Zbl

[7] A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi) groups”, European J. Math., 4:4 (2018), 1476–1490 | DOI | MR | Zbl

[8] S. Kerov, A. Vershik, “The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the $K_0$-functor theory of AF-algebras”, Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., 7, Gordon and Breach, 1990, 36–114 | MR

[9] J. F. C. Kingman, “Random partitions in population genetics”, Proc. Roy. Soc. London A, 361 (1978), 1–20 | DOI | MR | Zbl

[10] I. G. Macdonald, Symmetric Functions and Hall Folynomials, 2nd edition, Clarendon Press, Oxford, 1995 | MR

[11] R. P. Stanley, Enumerative Combinatorics, v. 2, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl