Scaling Limit of the Prudent Walk

Abstract : We describe the scaling limit of the nearest neighbour prudent walk on the square lattice, which performs steps uniformly in directions in which it does not see sites already visited. We show that the process eventually settles in one of the quadrants, and derive its scaling limit, which can be expressed in terms of a pair of independent stable subordinators. We also show that the asymptotic speed of the walk is well-defined in the L_1 -norm and equals 3/7. This process possesses unusual properties: it is ballistic but does not have an asymptotic direction, and several natural observables display ageing.
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Vincent Beffara, S. Friedli, Y. Velenik. Scaling Limit of the Prudent Walk. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2010, 15, pp.44 - 58. ⟨10.1214/ECP.v15-1527⟩. ⟨ensl-00364428⟩

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