Site monotonicity and uniform positivity for interacting random walks and the spin \(O(N)\) model with arbitrary \(N\). (English) Zbl 1445.60075
Summary: We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin \(O(N)\) model on the torus of \(\mathbb{Z}^d\), \(d\geq 3\), when \(N\in\mathbb{N}_{>0}\) and the inverse temperature \(\beta\) is large enough. This is a new result when \(N>2\) and extends the classical result of J. Fröhlich et al. [“Infrared bounds, phase transitions and continuous symmetry breaking”, Commun. Math. Phys. 50, 79–95 (1976)]. Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin \(O(N)\) model with arbitrary \(N \in \mathbb{N}_{>0}\), but for a wide class of systems of interacting random walks and loops, including the loop \(O(N)\) model, random lattice permutations, the dimer model, the double-dimer model, and the loop representation of the classical spin \(O(N)\) model.
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 |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
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