We consider periodic TASEP with periodic step initial condition, and evaluate the joint distribution of the locations of m particles. For arbitrary indices and times, we find a formula for the multi-time, multi-space joint distribution in terms of an integral of a Fredholm determinant. We then discuss the large time limit in the so-called relaxation scale. The one-point distributions for other initial conditions are also going to discussed.
Based on joint work with Zhipeng Liu (NYU).[-]

In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the understanding of the structure of the stationary non equilibrium states. I will discuss these problems in specific examples and from two different perspectives. The stochastic microscopic and combinatorial point of view and the macroscopic variational approach where the microscopic details of the models are encoded just by the transport coefficients.[-]

In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion process (TASEP). Then I will present a Fredholm determinant formula for its distribution (at a fixed time) and show how all the main properties of the fixed point (including the Markov property, space and time regularity, symmetries and scaling invariance, and variational formulas) can be derived from the formula and the construction, and also how the formula reproduces known self-similar solutions such as the $Airy_1andAiry_2$ processes.
The second part of the course will be devoted to explaining how the KPZ fixed point can be computed starting from TASEP. The method is based on solving, for any initial condition, the biorthogonal ensemble representation for TASEP found by Sasamoto '05 and Borodin-Ferrari-Prähofer-Sasamoto '07. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.
Based on joint work with K. Matetski and J. Quastel.[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will discuss an infinite dimensional Hamilton-Jacobi equation for the quasi-potential of stationary non equilibrium states, fluctuation-dissipation relationships, the underlying Hamiltonian structure, a relation with work and Clausius inequality, a large deviations functional for the current flowing through a system.[-]