Periodic TASEP with general initial conditions
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Periodic TASEP with general initial conditions Jinho Baik1 · Zhipeng Liu2 Received: 3 January 2020 / Revised: 17 July 2020 / Accepted: 13 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 largetime, 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. Keywords Periodic TASEP · Multi-point distribution · General initial condition · Kardar–Parisi–Zhang Universality Mathematics Subject Classification 60K35 · 82C22 · 60K37
1 Introduction In the last two decades, many limit theorems for the height function of the interacting particle systems in the Kardar–Parisi–Zhang (KPZ) universality class were established when the domain is infinite [1,7–9,19,24–26,28,30,31,38,39] or half-infinite [5,6,15, 37]. Recently, similar theorems were also beginning to be obtained for the case of the periodic domain. When the domain is periodic, all particles are strongly correlated
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Zhipeng Liu [email protected] Jinho Baik [email protected]
1
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
2
Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
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J. Baik, Z. Liu
if the period is too small compared with the time. Hence, it is natural to study the situation when the time t and the period L both become large and all particles are critically correlated. The critical case occurs when t = O(L 3/2 ), which is called the relaxation time scale. In this scale, the spatial correlation length and the period of the domain are of the same order of magnitude. One of the fundamental models in the KPZ universality class is the totally asymmetric simple exclusion process. We call the process in a periodic domain (equivalently the spatially periodic process) the periodic TASEP or PTASEP for short. On the other hand, we call the process on the infinite domain simply the TASEP. For the PTASEP, a few limit theorems in the relaxation time scale were obtained in the past three years.1 The papers [11,29,34] evaluated the limit of the one-point distribution function for three initial conditions; step, flat, and uniformly random initial conditions. For the step initial condition, the result was further extended to multi-point distributions at non-equal times in [12]. The goal of this paper is to
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