On the Decay of Correlations in Non-Analytic SO(n)-Symmetric Models

Naddaf, Ali

doi:10.1007/s002200050064

Cited by 6 publications

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“…Building on [4], Shlosman managed to obtain upper bounds of the same type for a much larger class of interactions [26], under some smoothness assumption on V . A similar, but less general, result was later obtained by Naddaf [22], using an adaptation of the McBryan-Spencer approach. More recently, Ioffe, Shlosman and Velenik showed how to dispense with the smoothness assumption, extending Shlosman's result to very general interactions V [11].…”

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confidence: 65%

Upper Bound on the Decay of Correlations in a General Class of O(N)-Symmetric Models

Gagnebin

Velenik

2014

Commun. Math. Phys.

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Abstract:We consider a general class of two-dimensional spin systems, with continuous but not necessarily smooth, possibly long-range, O(N )-symmetric interactions, for which we establish algebraically decaying upper bounds on spinspin correlations under all infinite-volume Gibbs measures.As a by-product, we also obtain estimates on the effective resistance of a (possibly long-range) resistor network in which randomly selected edges are shorted.

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confidence: 65%

Upper Bound on the Decay of Correlations in a General Class of O(N)-Symmetric Models

Gagnebin

Velenik

2014

Commun. Math. Phys.

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“…Perhaps surprisingly, the behavior of the two-dimensional model when n ≥ 2, so that the spin space S n−1 has a continuous symmetry, is quite different from that of the Ising model. The Mermin-Wagner theorem [89,88] asserts that in this case there is no phase with long-range order at any inverse temperature β. Quantifying the rate at which correlations decay has been the focus of much research along the years [69,75,101,36,106,107,99,109,54,73,21,90,92,72,57] and is still not completely understood. Improving on earlier bounds, McBryan and Spencer [86] showed in 1977 that the decay occurs at least at a power-law rate,…”

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confidence: 99%

Lectures on the Spin and Loop O(n) Models

Peled

Spinka

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the (n − 1)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (n = 1), the XY model (n = 2) and the Heisenberg model (n = 3). We discuss questions of longrange order (spontaneous magnetization) and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the number of spin components n. Among the topics presented are the Mermin-Wagner theorem, the Berezinskii-Kosterlitz-Thouless transition, the infra-red bound and Polyakov's conjecture on the two-dimensional Heisenberg model. 2The loop O(n) model is a model for a random configuration of disjoint loops. In these notes we discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight n ≥ 0 and an edge weight x ≥ 0. Special cases include self-avoiding walk (n = 0), the Ising model (n = 1), critical percolation (n = x = 1), dimer model (n = 1, x = ∞), proper 4-coloring (n = 2, x = ∞), integer-valued (n = 2) and tree-valued (integer n >= 3) Lipschitz functions and the hard hexagon model (n = ∞). The object of study in the model is the typical structure of loops. We will review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining. We then elaborate on recent results for the self-avoiding walk case and for large values of n.

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