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End date: 2022 × Start date: 2020 × Start date: 1984 × DDC: 510 × Type: Text × Type: report × Subject: phase transitions × Subject: random loop models ×

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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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    Macroscopic loops in the Bose gas, Spin O(N) and related models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Quitmann, Alexandra; Taggi, Lorenzo
    We consider a general system of interacting random loops which includes several models of interest, such as the textitSpin O(N) model, textitrandom lattice permutations, a version of the textitinteracting Bose gas in discrete space and of the textitloop O(N) model. We consider the system in ℤd, d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate ℤd by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.