Conformal walks in two dimensions

My thesis is devoted to the study of the critical properties of interacting walks in two dimensions, with particular emphasis on their conformal invariance. Two lattice models have been considered, which belong to different universality classes. The first is the well-known self-avoiding walk (SAW), which models polymers in good solutions. The second is the interacting self-avoiding walk, in which nearest-neighbor contacts between non-consecutive monomers are energetically favored. As the strength of this interaction is varied a (tri-)critical point is reached, called the $\theta$ point. This model describes polymers in poor solutions (under the critical temperature) and the critical transition between the latter and the swelled SAW-like behavior above the critical temperature. I first focused on the critical properties at the $\theta$ point, checking them against the theoretical predictions. I performed extensive Montecarlo simulations --- using an algorithm that implements both bi-local and non-local moves - and obtained high precision estimates for the critical exponents and the critical temperature, as well as for several quantities such as the universal ratios and the CSCPS expression. In particular, I studied the end-to-end distribution function thoroughly (both the short- and large-distance behaviors, and the phenomenological expression). Then I moved on to considering conformal-invariance related questions. The recently-introduced and widely studied Schramm-Loewner evolutions (SLE) are the perfect candidates for the scaling limits of conformally invariant walk models. SLE is a one-parameter family ($\kappa>0$) of random processes on the complex plane that produce random curves. Connection to the self-avoiding walk ($\kappa=8/3$) and to the $\theta$-point interacting self-avoiding walk ($\kappa=6$) has been established in the literature only in the half plane (the latter relies on the supposed equivalence with the critical percolation explorer). I checked this connection in the topologically different whole-plane case, focusing in particular on the distribution functions. To this aim, I devised a way to simulate SLE in $\mathbb C$ and introduced and studied some techniques to obtain the correct parametrization. I found that the distribution function of an inner point in the self-avoiding walk coincides with the end-point distribution function of whole-plane $\kappa=8/3$ SLE with the correct parametrization, thus providing numerical evidence in favor of a conjecture by Werner. Moreover, computing the position of an internal point in a SAW by exactly sampling the discrete SLE process turns out to be an efficient algorithm, which is open to further improvement. I did the same check for $\theta$-point polymers and $\kappa=6$ whole-plane SLE, and surprisingly found that the two distribution functions do not match. This unexpected result deserves further investigation, since it could unveil new aspects of the conformal invariance of $\theta$-point self-avoiding walks, and shed new light on the connection with critical percolation.