Reunion of random walkers with a long range interaction: applications to polymers and quantum mechanics

Rigidity of melting DNA

The temperature dependence of DNA flexibility is studied in the presence of stretching and unzipping forces. Two classes of models are considered. In one case the origin of elasticity is entropic due to the polymeric correlations, and in the other the double-stranded DNA is taken to have an intrinsic rigidity for bending. In both cases single strands are completely flexible. The change in the elastic constant for the flexible case due to thermally generated bubbles is obtained exactly. For the case of intrinsic rigidity, the elastic constant is found to be proportional to the square root of the bubble number fluctuation.

Efimov-like phase of a three-stranded DNA and the renormalization-group limit cycle

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.

Flory theory for polymers

We review various simple analytical theories for homopolymers within a unified framework. The common guideline of our approach is the Flory theory, and its various avatars, with the attempt at being reasonably self-contained. We expect this review to be useful as an introduction to the topic at the graduate student level.

Renormalization group limit cycle for three-stranded DNA

We show that there exists an Efimov-like three strand DNA bound state at the duplex melting point and it is described by a renormalization group limit cycle. A nonperturbative renormalization group is used to obtain this result in a model involving short range pairing only. Our results suggest that Efimov physics can be tested in polymeric systems.

Vicious walks with long-range interactions

The asymptotic behavior of the survival or reunion probability of vicious walks with short-range interactions is generally well studied. In many realistic processes, however, walks interact with a long-ranged potential that decays in d dimensions with distance r as r(-d-σ). We employ methods of renormalized field theory to study the effect of such long-range interactions. We calculate the exponents describing the decay of the survival probability for all values of parameters σ and d to first order in the double expansion in ε=2-d and δ=2-d-σ. We show that there are several regions in the σ-d plane corresponding to different scalings for survival and reunion probabilities. Furthermore, we calculate the leading logarithmic corrections.