Thordur Jonsson and Sigurður Örn Stefánsson 2009 J. Phys. A: Math. Theor. 42 485006 doi:10.1088/1751-8113/42/48/485006
Thordur Jonsson and Sigurður Örn Stefánsson
Show affiliationsWe study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph (the spine) to which leaves are attached. We prove that the model has two phases, a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability 1 and a single vertex of infinite order appears in the thermodynamic limit. We calculate the spectral dimension of the graphs in both phases and prove the existence of a Gibbs measure. We discuss generalizations of this model and the relationship with models of nongeneric random trees.
05.40.Fb Random walks and Levy flights
05.70.Jk Critical point phenomena
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
Issue 48 (4 December 2009)
Received 26 August 2009, in final form 20 October 2009
Published 17 November 2009
Thordur Jonsson and Sigurður Örn Stefánsson 2009 J. Phys. A: Math. Theor. 42 485006
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