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The boson gas on a Cayley tree. (English) Zbl 0893.60094
Summary: We analyze the free boson gas on a Cayley tree using two alternative methods. The spectrum of the lattice Laplacian on a finite tree is obtained using a direct iterative method for solving the associated characteristic equation and also using a random walk representation for the corresponding fermion lattice gas. The existence of the thermodynamic limit for the pressure of the boson lattice gas is proven and it is shown that the model exhibits boson condensation into the ground state. The random walk representation is also used to derive an expression for the Bethe approximation to the infinite-volume spectrum. This spectrum turns out to be continuous instead of a dense point spectrum, but there is still boson condensation in this approximation.

MSC:
60K40 Other physical applications of random processes
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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