Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps.

We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, which corresponds to the limit N-->0 of an N-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed by using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions for the critical crossover functions, finding good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal crossover behavior of our data for any finite range.

Simple model for the DNA denaturation transition.

We study pairs of interacting self-avoiding walks ¿omega(1), omega(2)¿ on the 3d simple cubic lattice. They have a common origin omega(1)(0)=omega 20, and are allowed to overlap only at the same monomer position along the chain: omega(1)(i) not equal omega(2)(j) for i not equal j, while omega(1)(i)=omega(2)(i) is allowed. The latter overlaps are indeed favored by an energetic gain epsilon. This is inspired by a model introduced long ago by Poland and Sheraga [J. Chem. Phys. 45, 1464 (1966)] for the denaturation transition in DNA where, however, self avoidance was not fully taken into account. For both models, there exists a temperature T(m) above which the entropic advantage to open up overcomes the energy gained by forming tightly bound two-stranded structures. Numerical simulations of our model indicate that the transition is of first order (the energy density is discontinuous), but the analog of the surface tension vanishes and the scaling laws near the transition point are exactly those of a second-order transition with crossover exponent straight phi=1. Numerical and exact analytic results show that the transition is second order in modified models where the self-avoidance is partially or completely neglected.