Geometrical properties of two-dimensional interacting self-avoiding walks at the θ-point

We perform a Monte Carlo simulation of two-dimensional N-step interacting self-avoiding walks at the θ-point, with lengths up to N = 3200. We compute the critical exponents, verifying the Coulomb-gas predictions, the θ-point temperature Tθ = 1.4986(11), and several invariant size ratios. Then, we focus on the geometrical features of the walks, computing the instantaneous shape ratios, the average asphericity, and the end-to-end distribution function. For the latter quantity, we verify in detail the theoretical predictions for its small- and large-distance behaviour.