arXiv 1702.028872017-02-09math-phmath.MPmath.PR

弱场中重正化的Dyson模型

Dyson models under renormalization and in weak fields

R. Bissacot, E. O. Endo, A. C. D. van Enter, B. Kimura, A. Le Ny, W. M. Ruszel

备注：: 15 pages. arXiv admin note: text overlap with arXiv:1603.05409

摘要：

我们考虑一维长程自旋模型（通常称为戴森模型），由伊辛铁磁体与缓慢衰减的长程对势的形式$\frac{1}{|i-j|^α}$主要集中在缓慢衰减的范围$1 < α\leq 2$。我们描述两个最近的结果，一个关于重整化和一个关于在低温外场的影响。第一个结果指出，在一维中的抽取远程吉布斯测度成为非吉布斯的，在相同的静脉中，在更高的维度的短程模型的可比结果。第二个结果解决了这种模型下的非均匀领域，特别是外部领域的多项式衰减为零的$(|i|+1)^{- γ}$的行为。我们研究了如何临界衰减功率的字段，$γ$，其中的相变持续和衰减功率$α$的戴森模型比较，扩展最近的结果为短程模型的格子和树上。我们还简要地指出这些结果之间的一些类比。

英文摘要：

We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form $\frac{1}{|i-j|^α}$ mainly focusing on the range of slow decays $1 < α\leq 2$. We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as $(|i|+1)^{- γ}$. We study how the critical decay power of the field, $γ$, for which the phase transition persists and the decay power $α$ of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.

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摘要：

英文摘要：

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