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We prove large deviations principles (LDPs) for the perimeter and the area ofthe convex hull of a planar random walk with finite Laplace transform of itsincrements.  We give explicit upper and lower bounds for the rate function of theperimeter in terms of the rate function of the increments. These boundscoincide and thus give the rate function for a wide class of distributionswhich includes the Gaussians and the rotationally invariant ones. For randomwalks with such increments, large deviations of the perimeter are attained bythe trajectories that asymptotically align into line segments. However, linesegments may not be optimal in general.  Furthermore, we find explicitly the rate function of the area of the convexhull for random walks with rotationally invariant distribution of increments.For such walks, which necessarily have zero mean, large deviations of the areaare attained by the trajectories that asymptotically align into half-circles.For random walks with non-zero mean increments, we find the rate function ofthe area for Gaussian walks with drift. Here the optimal limit shapes areelliptic arcs if the covariance matrix of increments is non-degenerate andparabolic arcs if otherwise.  The above results on convex hulls of Gaussian random walks remain valid forconvex hulls of planar Brownian motions of all possible parameters. Moreover,we extend the LDPs for the perimeter and the area of convex hulls to generalLevy processes with finite Laplace transform.

Large deviations of convex hulls of planar random walks and Brownian motions