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We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion(More)
We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behaviour and they are also sensitive to the initial(More)
We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability L N (t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability R N (t) that the rightmost particle never becomes the leftmost. We(More)
A diffusion-limited aggregation process, in which clusters coalesce by means of 3-particle reaction, A + A + A → A, is investigated. In one dimension we give a heuristic argument that predicts logarithmic corrections to the mean-field asymptotic behavior for the concentration of clusters of mass m at time t, C m (t) ∼ m −1/2 (log(t)/t) 3/4 , for 1 ≪ m ≪ t/(More)
We study a class of directed random graphs. In these graphs, the interval [0, x] is the vertex set, and from each y ∈ [0, x], directed links are drawn to points in the interval (y, x] which are chosen uniformly with density one. We analyze the length of the longest directed path starting from the origin. In the x → ∞ limit, we employ traveling wave(More)
We investigate choice-driven network growth. In this model, nodes are added one by one according to the following procedure: for each addition event a set of target nodes is selected, each according to linear preferential attachment, and a new node attaches to the target with the highest degree. Depending on precise details of the attachment rule, the(More)
Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution(More)