The Critical Attractive Random Polymer in Dimension One

van der Hofstad, Remco; Klenke, Achim; Konig, Wolfgang

HPL-BRIMS-2001-05

Keyword(s): one-dimensional polymers; repulsive and attractive interaction; phase transition; local times of simple random walk

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. A polymer chain with attractive and repulsive forces between the monomers is modeled by attaching a weight e-* for every self-intersection and e /(2d) for every self-contact to the probability of an n-stepsimple random walk on Z d , where *, > 0 are parameters. It is known that for d =1 and > * the chain collapses down to finitely many sites, while for d =1 and <* it spreads out ballistically. Here we study for d = 1 the critical case =* and show that the end- to-end distance runs on the scale n = (log n) -1/4 . We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size n times a constant. Moreover, we derive the asymptotics of the partition function. Notes: Remco van der Hofstad, Department of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands. Achim Klenke, Department of Mathematics, Erlangen-Nurnberg University, Bismarckstrae 1 , 91054 Erlangen, Germany

30 Pages

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