The critical attractive random polymer in dimension one

摘要

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e(-beta) for every self-intersection and e(gamma/(2d)) for every self-contact to the probability of an n-step simple random walk on Z(d), where beta, gamma > 0 are parameters. It is known that for d = 1 and gamma > beta the chain collapses down to finitely many sites, while for d = 1 and gamma < beta it spreads out ballistically. Here we study for d = 1 the critical case gamma = beta corresponding to the collapse transition and show that the end-to-end distance runs on the scale alpha(n) = rootn (log n)(-1/4). We describe the asymptotic shape of them accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size a. times a constant. Moreover, we derive the asymptotics of the partition function. [References: 8]