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Register a new userMulti-point distribution of TASEP
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Zhipeng Liu
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Recently Johansson and Rahman obtained the limiting multi-time distribution for the discrete polynuclear growth model, which is equivalent to discrete TASEP model with step initial condition. In this paper, we obtain a finite time multi-point distribution formula of continuous TASEP with general initial conditions in the space-time plane. We evaluate the limit of this distribution function when the times go to infinity proportionally for both step and flat initial conditions. These limiting distributions are expected to be universal for all the models in the Kardar-Parisi-Zhang universality class.
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The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.
We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles rho. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.
In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of Z, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.
We provide a direct and elementary proof that the formula obtained in [MQR17] for the TASEP transition probabilities for general (one-sided) initial data solves the Kolmogorov backward equation. The same method yields the solution for the related PushASEP particle system.
We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and step-flat initial conditions satisfy. Hence, we obtain the limit theorem for these three initial conditions in the periodic model, extending the previous work on the step initial condition. We also consider uniform random and uniform-step random initial conditions.
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