Interacting self-avoiding polygons

Betz, Volker; Schäfer, Helge; Taggi, Lorenzo

doi:10.1214/19-aihp1003

Cited by 8 publications

(6 citation statements)

References 9 publications

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“…For any pair of distinct vertices x, y ∈ V, let Ω per (x, y) be the set of functions π : V \ {y} → V \ {x} such that, for every z ∈ V, either {z, π(z)} ∈ E or π(z) = z, and, moreover, every z ∈ V \ {x, y} has precisely one input and one output in π (from this it also follows that x has precisely one output and that y has precisely one input). This model has been studied in [1,2,3].…”

Section: Random Lattice Permutationsmentioning

confidence: 99%

Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

Lees

Taggi

2019

Commun. Math. Phys.

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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 Z d , d ≥ 3, when N ∈ N >0 and the inverse temperature β is large enough. This is a new result when N > 2 and extends the classical result of Fröhlich, Simon and Spencer (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 ∈ 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. *

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Section: Random Lattice Permutationsmentioning

confidence: 99%

Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

Lees

Taggi

2019

Commun. Math. Phys.

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“…A first important progress was made in [2,16] (see also [25,Chapter 11]), where the occurrence of macroscopic loops was proved for the Bose gas under hard-core local interactions under the so-called half-filling condition. Further important progress has been made in the rigorous analysis of spatial random permutation models [5,7,8,9,11,19] and in the Bosonic loop soup considered in [15]. In these systems the loops interact through a potential which depends on the total number of loops of a given length.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Macroscopic loops in the Bose gas, Spin O(N) and related models

Quitmann¹,

Taggi²

2022

Preprint

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We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in Z d , d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate Z d 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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“…Additionally, our proof is also quite flexible and, for example, it holds for any graph of bounded degree; it holds on Z d with finite range (not necessarily translation invariant) coupling constants, and it holds for a class of models with continuous symmetry whose interaction does not necessarily take the form e −H (with H representing the hamiltonian function) -these models are 'less physical' but they lead to interesting random loop models, for example the loop O(N) model [8,9,16,19] (see Sect. 5.2), see also [3,5] for related models.…”

Section: Introductionmentioning

confidence: 98%

Exponential decay of transverse correlations for O(N) spin systems and related models

Lees

Taggi

2021

Probab. Theory Relat. Fields

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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 $$N > 1$$ N > 1 . Our result is new when $$N > 3$$ N > 3 , in which case no Lee–Yang theorem is available, it is an alternative to Lee–Yang when $$N = 2, 3$$ 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.

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Interacting self-avoiding polygons

Cited by 8 publications

References 9 publications

Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

Macroscopic loops in the Bose gas, Spin O(N) and related models

Exponential decay of transverse correlations for O(N) spin systems and related models

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