Interacting self-avoiding polygons

Betz, Volker; Schäfer, Helge; Taggi, Lorenzo

doi:10.48550/arxiv.1805.08517

Cited by 2 publications

(2 citation statements)

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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: Special Casesmentioning

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: Special Casesmentioning

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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“…Our random wire model is an extension of the random current representation of the Ising model that was introduced by Griffiths, Hurst, and Sherman [24], and popularised by Aizenman [1]. It is also related to the Brydges-Fröhlich-Spencer representation of spin O(N ) models [15,19], and to loop O(N ) models [30,9]. The Poisson-Dirichlet distribution of random partitions was introduced by Kingman [27]; it is the invariant measure for the split-merge (coagulation-fragmentation) process [35,16,7].…”

Section: Introductionmentioning

confidence: 99%

Loop Correlations in Random Wire Models

Benassi

Ueltschi

2019

Commun. Math. Phys.

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We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson-Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson-Dirichlet counterpart.1991 Mathematics Subject Classification. 60C05, 60K35, 82B05, 82B20, 82B26.

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Interacting self-avoiding polygons

Cited by 2 publications

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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

Loop Correlations in Random Wire Models

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