We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter L and the mean area A have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P (A) and P (L). In this paper, we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P (A) and P (L) are as small as 10 −300 . We analyse (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T , a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/ √ T , respecively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

We study the polygons governing the convex hull of a point set created by the steps of n independent two-dimensional random walkers. Each such walk consists of T discrete time steps, where x and y increments are i.i.d. Gaussian. We analyze area A and perimeter L of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below 10 −900 . We find that the densities exhibit a universal scaling behavior as a function of A/T and L/ √ T , respectively. As in the case of one walker (n = 1), the densities follow Gaussian distributions for L and √ A, respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for T → ∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n → ∞ as found in the n = 1 case. We also investigated the behavior of the averages as a function of the number of walks n and found good agreement with the predicted behavior.

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We consider the negative weight percolation (NWP) problem on hypercubic lattice graphs with fully periodic boundary conditions in all relevant dimensions from d=2 to the upper critical dimension d=6. The problem exhibits edge weights drawn from disorder distributions that allow for weights of either sign. We are interested in the full ensemble of loops with negative weight, i.e., nontrivial (system spanning) loops as well as topologically trivial ("small") loops. The NWP phenomenon refers to the disorder driven proliferation of system spanning loops of total negative weight. While previous studies where focused on the latter loops, we here put under scrutiny the ensemble of small loops. Our aim is to characterize-using this extensive and exhaustive numerical study-the loop length distribution of the small loops right at and below the critical point of the hypercubic setups by means of two independent critical exponents. These can further be related to the results of previous finite-size scaling analyses carried out for the system spanning loops. For the numerical simulations, we employed a mapping of the NWP model to a combinatorial optimization problem that can be solved exactly by using sophisticated matching algorithms. This allowed us to study here numerically exact very large systems with high statistics.