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Cube root fluctuations for the corner growth model associated to the exclusion process. (English) Zbl 1139.60046
The authors are interested in the last-passage growth model on the planar integer lattice, i.e. the maximal sum of exponential random weights along up-right paths in the first quadrant. The interior weights have rate 1, and the boundary weights on the axes have rate \(1-\rho\) and \(\rho\). The variance of the last-passage time is related to the point where the maximal path exits the axes and proved to be of order \(t^{2/3}\) in a characteristic direction. By decreasing the weights on the axes in an arbitrary way, the fluctuations of the last-passage time are proved to be still of order \(t^{1/3}\). This includes the situation known as the rarefaction fan. The proof is an adaptation of a recent work of E. Cator and P. Groeneboom [Ann. Probab. 34, No. 4, 1273–1295 (2006; Zbl 1101.60076)] concerning Hammersley’s process and utilizes the competition interface introduced by P. A. Ferrari, J. B. Martin and Pimentel [Phys. Rev. E., 73:031602 (2006)]. But here the arguments are entirely probabilistic.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
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