Two-dimensional wetting with binary disorder: a numerical study of the loop statistics

摘要

We numerically study the wetting (adsorption) transition of a polymer chainon a disordered substrate in 1+1 dimension.Following the Poland-Scheraga modelof DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops.This allows us to consider chain lengths of order $N \sim 10^5 $ to $10^6$,with $10^4$ disorder realizations. Our study is based on the statistics ofloops between two contacts with the substrate, from which we define Binder-likeparameters: their crossings for various sizes $N$ allow a precise determinationof the critical temperature, and their finite size properties yields acrossover exponent $\phi=1/(2-\alpha) \simeq 0.5$.We then analyse atcriticality the distribution of loop length $l$ in both regimes $l \sim O(N)$and $1 \ll l \ll N$, as well as the finite-size properties of the contactdensity and energy. Our conclusion is that the critical exponents for thethermodynamics are the same as those of the pure case, except for stronglogarithmic corrections to scaling. The presence of these logarithmiccorrections in the thermodynamics is related to a disorder-dependentlogarithmic singularity that appears in the critical loop distribution in therescaled variable $\lambda=l/N$ as $\lambda \to 1$.