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In this chapter, we study a certain type of \(C^{*}\) -probability spaces induced by connected finite directed graphs. We are interested in free product of such \(C^{*}\) -probability spaces obtained by a combinatorial process constructing new graphs from given multigraphs under a rule, called the gluing. We characterize operator-algebraic, and free-probabilistic structures of resulted \(C^{*}\) -probability spaces from the gluing in terms of (free-probability-theoretic) free product (also implying the combinatorial properties of graphs), and consider free-distributional data on them. In particular, we are interested in semicircular elements whose free distributions are the semicircular law.

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Vertex-Glued Connected Finite Graphs and Mutually Free Semicircular Elements

  • 会议论文

  • 发表日期：

    2026年

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  • Ilwoo Cho,
  • Palle E. T. Jorgensen

摘要

In this chapter, we study a certain type of \(C^{*}\) -probability spaces induced by connected finite directed graphs. We are interested in free product of such \(C^{*}\) -probability spaces obtained by a combinatorial process constructing new graphs from given multigraphs under a rule, called the gluing. We characterize operator-algebraic, and free-probabilistic structures of resulted \(C^{*}\) -probability spaces from the gluing in terms of (free-probability-theoretic) free product (also implying the combinatorial properties of graphs), and consider free-distributional data on them. In particular, we are interested in semicircular elements whose free distributions are the semicircular law.