2018
DOI: 10.1214/18-ecp189
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Shifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$

Abstract: In the loop O(n) model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional towhere λ, n ∈ [0, ∞). Let µ be the connective constant of the lattice and, for any n ∈ [0, ∞), let λ c (n) be the largest value of λ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that λ c (n) = 1/µ when n = 0 (in this case the model corresponds to the self-avoiding walk) and that for any n ≥ 0, λ c (… Show more

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Cited by 9 publications
(8 citation statements)
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“…To obtain the loop O(N ) model which was defined in [7], one would need to define the weight function U slightly differently than (2.10). See also [8,9,16,17,32] for recent papers.…”
Section: Special Casesmentioning
confidence: 99%
“…To obtain the loop O(N ) model which was defined in [7], one would need to define the weight function U slightly differently than (2.10). See also [8,9,16,17,32] for recent papers.…”
Section: Special Casesmentioning
confidence: 99%
“…On the contrary, if λ is sufficiently small the model exhibits a quite different behaviour, indeed E L,U,v,N,λ (|Γ x |) = O(1) in the limit as L → ∞ and the quantity in the left-hand side of (1.12) decays exponentially with the distance between x and y (with exponential moments uniformly bounded in L). This can be proved using the cluster expansion method (see for example [21,Chapter 5]) or the methods of [6,36]. Hence, the combination of these facts and of our theorem imply the occurrence of a phase transition with respect to the variation of the parameter λ.…”
Section: Main Results About the Occurrence Of Macroscopic Loopsmentioning
confidence: 66%
“…Lastly, it is proved there that when n = 1 and 1 < x ≤ √ 3 (antiferromagnetic Ising) the model has loops of large diameter (comparable to that of the domain) with nonnegligible probability. On the other side, Taggi [116] established exponential decay of loop lengths when n > 0 and x ≤ ( 2 + √ 2) −1 + ε(n), with ε(n) > 0 some function of n. Glazman-Manolescu [61] further showed exponential decay for any n > 1 and x < 1 n = x = 1, and showing the existence of large loops in the remaining parts of the phase diagram: for 0 < n < 1 and any x, or for n = 1 and any x ∈ ( √ 3, ∞] (it is unknown even for the dimer model case, x = ∞), or for 1 < n ≤ 2 and x > x c (n) (apart from the case n = 2, x = 1 and from the neighborhood of n = x = 1 mentioned above) .…”
Section: Conjectured Phase Diagram and Rigorous Resultsmentioning
confidence: 99%