A note on the asymptotics of the polymer measures in one dimension (Q1210214)

Based on \textit{J. Westwater}'s work [Trends and developments in the eighties, Bielefeld Encounters Math. Phys. 4 and 5, 384-404 (1985; Zbl 0583.60066)], it is further proved that \[ \lim_{T\to\infty}E_{\nu(g,T)}| w(T)/T|^ \alpha=(g^{1/3}C)^ \alpha, \] \(\forall\alpha>0\), for some constant \(C>0\), where \(\nu(g,T)\) is the one-dimensional polymer measure defined on \(C_ 0([0,T]\to R^ 1)\), and \(\{w(t)\}\), \(t\in[0,T]\), is the corresponding coordinate process of the Wiener process. It should be noted that this satisfactory result is only for \(d=1\), and the conjecture for the asymptotic behavior of the higher dimensional polymer measures is still unproved or unknown.