Abstract
We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application, we derive site-monotonicity properties for the spin–spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates—improving previous positivity results which hold for the Cesàro sum. We also derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model and lattice permutations, thus extending the previous results of Lees and Taggi (2019).
Highlights
We consider a general probabilistic model on the torus TL = Zd /LZd, whose realisations live in a product of local spaces
Each local space is associated to one of the vertices of TL and elements of the local spaces interact with each other via a linear functional acting on a real algebra of observables
If the linear functional acting on functions of our state space is reflection positive, several site-monotonicity properties for the two-point function hold
Summary
Each local space is associated to one of the vertices of TL and elements of the local spaces interact with each other via a linear functional acting on a real algebra of observables. This general setting includes various important models in statistical mechanics, for example the spin O(N) model, the quantum Heisenberg anti-ferromagnet and X Y model, the dimer and the double-dimer model, lattice permutations, and the loop O(N) model. If the linear functional acting on functions of our state space is reflection positive, several site-monotonicity properties for the two-point function hold
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