Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Rotor-router aggregation on the layered square lattice. | Rotor-router aggregation on the layered square lattice Wouter Kager VU University Amsterdam Department of Mathematics De Boelelaan 1081 1081HV Amsterdam The Netherlands wkager@ Lionel Levine Massachusetts Institute of Technology Department of Mathematics Cambridge MA 02139 USA levine@ Submitted Mar 31 2010 Accepted Oct 19 2010 Published Nov 5 2010 Mathematics Subject Classification 2010 82C24 Abstract In rotor-router aggregation on the square lattice z2 particles starting at the origin perform deterministic analogues of random walks until reaching an unoccupied site. The limiting shape of the cluster of occupied sites is a disk. We consider a small change to the routing mechanism for sites on the x- and y-axes resulting in a limiting shape which is a diamond instead of a disk. We show that for a certain choice of initial rotors the occupied cluster grows as a perfect diamond. 1 Introduction Recently there has been considerable interest in low-discrepancy deterministic analogues of random processes. An example is rotor-router walk PDDK96 a deterministic analogue of random walk. Based at every vertex of the square grid z2 is a rotor pointing to one of the four neighboring vertices. A chip starts at the origin and moves in discrete time steps according to the following rule. At each time step the rotor based at the location of the chip turns clockwise 90 degrees and the chip then moves to the neighbor to which that rotor points. Holroyd and Propp HP09 show that rotor-router walk captures the mean behavior of random walk in a variety of respects stationary measure hitting probabilities and hitting times. Cooper and Spencer CS06 study rotor-router walks in which n chips starting at arbitrary even vertices each take a fixed number t of steps showing that the final locations of the chips approximate the distribution of a random walk run for t steps to within constant error independent of n and t. Rotor-router walk and other low-discrepancy .
