Positive fixed points of cubic operators on \(\mathbb{R}^2\) and Gibbs measures. (English) Zbl 1534.60143
Summary: One model with nearest neighbour interactions of spins with values from the set \([0,1]\) on the Cayley tree of order three is considered in the paper. Translation-invariant Gibbs measures for the model are studied. Results are proved by using properties of the positive fixed points of a cubic operator in the cone \(\mathbb{R}_+^2 \).
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B05 | Classical equilibrium statistical mechanics (general) |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
Keywords:
Cayley tree; Gibbs measure; translation-invariant Gibbs measure; fixed point; cubic operator; Hammerstein’s integral operatorReferences:
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