A note on the asymptotics of the polymer measures in one dimension

Giveng∈(0, ∞), we prove that $$\mathop {\lim }\limits_{T \to \infty } E_{v(g,T)} \left| {\frac{{W(T)}}{T}} \right|^\alpha = (g^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} C)^\alpha ,\forall \alpha > 0$$ for some constantC∈(0, ∞), wherev(g, T) is the polymer measure defined onC ₀(0,T] →R ¹), and {W(t)} ^t∈0,T] is the corresponding coordinate process.