ON CONTINUOUS-TIME SELF-AVOIDING RANDOM-WALK IN DIMENSION 4

We study the distribution of the end-to-end distance of continuous-tim e self-avoiding random walks (CTRW) in dimension four from two viewpoi nts. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a wea kly self-avoiding random walk (SARW) that penalizes two-body interacti ons of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-di mensional integer Lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time ra ndom walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self -avoiding random walks.