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Start date: 1984 × Type: Text × Type: report × Subject: infrared bound × Subject: phase transitions ×

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    Site-monotonicity properties for reflection positive measures with applications to quantum spin systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lees, Benjamin; Taggi, Lorenzo
    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 of such a general theorem, we derive site-monotonicity properties for the spin-spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that such spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates -- improving previous positivity results which hold for the Cesàro sum -- and we 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, lattice permutations, thus extending the previous results of Lees and Taggi (2019).
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    Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $mathbbZ^d, d > 2$, via reflection positivity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Taggi, Lorenzo
    Our first main result is that correlations between monomers in the dimer model in ℤd do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.