Localization in the ground state of a disordered array of quantum rotators

We consider the zero-temperature behavior of a disordered array of quantum rotators given by the finite-volume Hamiltonian: $$H_Lambda = - mathop Sigma limits_{x in Lambda } frac{{h(x)}}{2}frac{{partial ^2 }}{{partial varphi (x)^2 }} - Jmathop Sigma limits_{leftlangle {x,y} rightrangle in Lambda } cos (varphi (x) - varphi (y))$$ , wherex,y∈Z ^d , 〈,〉 denotes nearest neighbors inZ ^d ;J>0 andh={h(x)>0,x∈Z ^d } are independent identically distributed random variables with common distributiondμ(h), satisfying ∫h ^?δ dμ(h)<∞ for some δ>0. We prove that for anym>0 it is possible to chooseJ(m) sufficiently small such that, if 0<J<J(m), for almost every choice ofh and everyx∈Z ^d the ground state correlation function satisfies $$leftlangle {cos (varphi (x) - varphi (y))} rightrangle leqq C_{x,h,J} e^{ - mleft| {x - y} right|} $$ for ally∈Z ^d withC _x,h,J <∞.