格点Z^3上临界超对称场论的整体重整化群轨线
The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
摘要：
我们考虑$Z^3$中的欧氏超对称场论，它由标量玻色子场和Grassmann场上的无质量高斯测度的超对称$Φ^4$扰动给出，协方差为$Z^3$中（稳定）Lévy随机游动的格林函数.格林函数依赖于Lévy-Khintchine参数$α={3+ε\over 2}$和$0<α<2$。对于$α={3\over 2}$，$Φ^{4}$的相互作用很小。我们证明$α-{3\over 2}={ε\over 2}>0$足够小，初始参数在一个适当的域中，存在一个全球性的重整化群轨迹一致有界的所有重整化群规模，因此在晶格上成为任意罚款。同时，我们建立了临界（稳定）流形的存在性。相互作用在所有尺度上都是一致有界的，因此我们正在构建一个所有尺度上的非高斯超对称场论。这个理论的兴趣来自于一个容易建立的事实，即$Z^3$中（弱）自回避Lévy行走的格林函数是控制这个模型的超对称测度的二阶矩（两点相关函数）。对重整化群轨道的控制是研究格林函数渐近性的准备。对临界重整化群轨道的严格控制为研究$Z^3$中（弱）自回避Lévy游动的临界指数作了准备.
英文摘要：
We consider an Euclidean supersymmetric field theory in $Z^3$ given by a supersymmetric $Φ^4$ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green's function of a (stable) Lévy random walk in $Z^3$. The Green's function depends on the Lévy-Khintchine parameter $α={3+ε\over 2}$ with $0<α<2$. For $α={3\over 2}$ the $Φ^{4}$ interaction is marginal. We prove for $α-{3\over 2}={ε\over 2}>0$ sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green's function of a (weakly) self-avoiding Lévy walk in $Z^3$ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The control of the renormalization group trajectory is a preparation for the study of the asymptotics of this Green's function. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in $Z^3$.
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