## One point fluctuations of periodic TASEP with step and flat initial conditions |

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Zhipeng Liu (NYU)Tue, Mar 1

We consider the periodic TASEP model on the space \{(x_1,\cdots,x_N)\in Z^N; x_1<\cdots<x_N<x_1+M\}. This model can be described as the TASEP on the ring with length M, or directed last passage percolation with periodic entries, or directed last passage percolation on a cylinder. We are interested in the fluctuations of a fixed particle as N, M, and time t all go to infinity. For the step initial condition, if the density of particles N/M is fixed, we prove that the limiting distribution is given by either GUE Tracy-Widom distribution or the product of two GUE Tracy-Widom distributions when t<N^{3/2}. We also find the explicit formula for the limit of the one point distribution in when t~N^{3/2}. For flat initial conditions, we have a similar formula.

This is a joint work with Jinho Baik.