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).[-]

The Spherical Sherrington-Kirkpatrick (SSK) model is defined by the Gibbs measure on a highdimensional sphere with a random Hamiltonian given by a symmetric quadratic function. The free energy at the zero temperature is the same as the largest eigenvalue of the random matrix associated with the quadratic function. Even for the finite temperature, there is a simple relationship between the free energy and the eigenvalues. We will discuss how one can study the fluctuations of the free energy using this relationship and results from random matrix theory. We will also discuss the distribution of the spin sampled from the Gibbs measure.[-]