We consider the weakly coupled $\phi^4 $ theory on $\mathbb Z^4 $, in a weak magnetic field $h$, and at the chemical potential $\nu_c $ for which the theory is critical if $h=0$. We prove that, as $h\to 0$, the magnetization of the model behaves as $(h\log h^{-1})^{\frac 13} $, and so exhibits a logarithmic correction to mean field scaling behavior. This result is well known to physicists, but had never been proven rigorously. Our proof uses the classic construction of the critical theory by Gawedzki and Kupiainen, and a cluster expansion with large blocks.

中文翻译：

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弱耦合 $$\phi _4^4 $$ϕ44 模型磁化的临界指数

我们考虑在 $\mathbb Z^4 $ 上的弱耦合 $\phi^4 $ 理论，在弱磁场 $h$ 和化学势 $\nu_c $ 下，如果 $h=0，该理论是关键的$. 我们证明，当 $h\to 0$ 时，模型的磁化表现为 $(h\log h^{-1})^{\frac 13} $，因此表现出对平均场缩放行为的对数校正. 这一结果为物理学家所熟知，但从未得到严格证明。我们的证明使用了 Gawedzki 和 Kupiainen 的批判理论的经典构造，以及具有大块的集群扩展。