Ergodicity of the Martin boundary of the Young--Fibonacci graph. II

Ergodicity of the Martin boundary of the Young--Fibonacci graph. I

Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $β\in(0,1]$ (the case $β=0$ corresponds to the Plancherel measure). In this article we prove the statements which correspond to the case $β=1$.

The Martin Boundary of the Young-Fibonacci Lattice

We find the Martin boundary for the Young-Fibonacci lattice YF. Along with the lattice of Young diagrams, this is the most interesting example of a differential poset. The Martin boundary construction provides an explicit Poisson-type integral representation of non-negative harmonic functions on YF. The latter are in a canonical correspondence with a set of traces on Okada locally semisimple algebra. The set is known to contain all the indecomposable traces. Presumably, all of the traces in the set are indecomposable, though we have no proof of this conjecture. Using a new explicit product for…

Young--Fibonacci 图的跳跃图的中心测度

Martin boundary of the jump graph on subgraphs of the Young--Fibonacci graph

本文研究 Young--Fibonacci 图中至多含 k 个 2 的单词构成的子图所对应的跳跃图，并完整刻画了该图路径空间上的所有中心测度。

Young–Fibonacci图的r-微分版本上的中心测度

The Martin boundary of the $r$-differential version of the Young--Fibonacci graph

本文刻画了Young–Fibonacci图r-微分版本路径空间的Martin边界，并证明了相应中心测度的遍历性。

The Mostar index of Fibonacci and Lucas cubes

The Mostar index of a graph was defined by Došlić, Martinjak, Škrekovski, Tipurić Spužević and Zubac in the context of the study of the properties of chemical graphs. It measures how far a given graph is from being distance-balanced. In this paper, we determine the Mostar index of two well-known families of graphs: Fibonacci cubes and Lucas cubes.