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Type: report × Subject: phase transitions × Subject: statistical mechanics ×

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Now showing 1 - 3 of 3
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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.
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    Macroscopic loops in the $3d$ double-dimer model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Quitmann, Alexandra; Taggi, Lorenzo
    The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in ℤ d, d>2, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [Betz, Taggi] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when d >2, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.
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    Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lees, Benjamin; Taggi, Lorenzo
    We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when N > 3, in which case no Lee-Yang theorem is available, it is an alternative to Lee-Yang when N = 2, 3, and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a `colour-switch' lemma, and a sampling procedure which allows us to bound from above the `typical' length of the open paths.