A Limit Theorem for the Wick Exponential of the Free Lattice Fields

#### S. Albeverio, S. Liang

2005, v.11, Issue 1, 157-164

ABSTRACT

Let $G_a$ be the free lattice field measure of mass $m_0$ on $a Z^2$, and let $:\exp\{\alpha \phi_x\}:$ be the corresponding Wick exponential of the lattice field $\phi_x$. Let $A \subset R^2$ be a bounded region and $a'(a) \ge a$ satisfy: $\lim_{a \to 0} a'(a) = 0$. In this paper, a limit theorem for the distribution of $a'^2 \sum_{x \in a' Z^2 \cap A} :\exp\{\alpha \phi_x\}:$ under $G_a$ is given, under the condition $\lim_{a \to 0} a'^4 |\log a| = \infty$. The corresponding problem for the $:\phi_x^4:$-field has been studied by Albeverio and Zhou [S. Albeverio and X.Y. Zhou, A central limit theorem for the fourth Wick power of the free lattice field, Commun. Math. Phys., 1996, v.181, no.1, 1-10].

Keywords: free lattice fields,continuum limit,asymptotics ofrandom fields,extreme events,i.i.d. random variables,quantum fields