One dimensional exclusion processes are stochastic processes in which many particles perform random walks under exclusion constraint. They have been playing important role in the fields of stochastic interacting systems in probability theory and non-equilibrium statistical mechanics in physics. For the last two decades, fluctuations of the processes have been studied quite intensively, since the seminal work by Johansson[1-1] on totally asymmetric simple exclusion process (TASEP) showing that the current fluctuation of TASEP with step initial condition is described by the GUE Tracy-Widom distribution. There have been a vast accumulation of generalizations and related results, but there are still many intriguing questions and problems to be solved.

In these lectures, we discuss a few new directions in the studies of fluctuations of exclusion processes. We also stress that such studies provide valuable insight to other methods based on hydrodynamic ideas which can be applied to a wider class of interacting particle systems. In the first lecture we review the basics of the subject. After introducing a few models such as the asymmetric simple exclusion process(ASEP) and the Kardar-Parisi-Zhang (KPZ) equation, we explain how one can study their fluctuations for the case of TASEP[1-2]. In the second lecture, we show that an approach introduced in [2] using Frobenius determinant can be applied to a large class of models in a unified manner. In the third lecture we explain our recent result on a two-species exclusion process and connection to the nonlinear fluctuating hydrodynamics[3]. In the last lecture we will consider an application of the techniques to study the large derivation in the symmetric exclusion process[4-1,2].

**References**

- [1-1] K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. (2009) 437-476. [arXiv:math/9903134]
- [1-2] T. Sasamoto, Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques, J. Stat. Mech. (2007) P07007. [arXiv:0705.2942]
- [2] T. Imamura, T. Sasamoto, Fluctuations for stationary q- TASEP, to appear in Prob. Th. Rel. Fields. [arXiv:1701.05991]
- [3] Z. Chen, J. de Gier, I. Hiki, T. Sasamoto, Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process, Phys. Rev. Lett. 120, 240601 (2018). [arXiv:1803.06829]
- [4-1] T. Imamura, K. Mallick, T. Sasamoto, Large deviations of a tracer in the symmetric exclusion process, Phys. Rev. Lett. 118, 160601 (2017). [arXiv:1701.05991]
- [4-2] T. Imamura, K. Mallick, T. Sasamoto, Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition, arXiv:1810.06131.

- Problem set 1 - up to (ii) •
**deadline - Monday, June 3, 11pm** - Problem set 1 solutions
- Problem set 2 - from (iii) on page 1, to the end of page 2 •
**deadline - Wednesday, June 5, 11pm** - Problem set 3 •
**deadline - Friday, June 7, 11pm**