We will reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. Through a one-dimensional irreversible diffusion process on torus, we explicit characterize essential concepts in the weak KAM theory such as the Peierls barrier, the projected Aubry/Mather sets, and the variational formulas for a self-consistent boundary data at each local attractors and for the rate function. The weak KAM representation of Freidlin-Wentzell's variational construction of the rate function is discussed based on the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton-Jacobi equation satisfying Freidlin-Wentzell's variational formula for the boundary data, which is the selected unique weak KAM solution and serves as the global energy landscape of the original stochastic process. The probability interpretations of the global energy landscape from the weak KAM perspective are also discussed.

This is a joint work with Yuan Gao of Purdue University.

Proteins and their interactions are the key component of cellular processes. It has been recently shown that the Artificial Intelligence solves virtually all protein structures, at accuracy equal to or greater than experiment. This is causing a tectonic shift in life sciences, both experimental and computational. This drastic paradigm change requires a major transition in computational methodology - from pattern recognition that for decades has been the driving force in computational biology, to stochastic modeling of live processes. Our study bridges two computational biology techniques - protein docking (based on pattern recognition) and kinetic simulations, developing an approach that can reach unprecedented simulation times at all-atom resolution. The energy landscape of a cell-size system was mapped by the pairwise Fast Fourier Transform docking and sampled by Markov Chain Monte Carlo protocol. The procedure was parametrized on existing data and validated on observations from experiments and molecular dynamics simulations. The simulation protocol performed consistently across very different systems of proteins at different concentrations. It recapitulated data on the previously observed diffusion rates and aggregation. The speed of calculation allows reaching unprecedented second-long trajectories of protein systems that approach the size of the cells (tens of millions of atoms), at atomic resolution. Such trajectories are tens-to-hundreds of thousands times longer than those modeled by standard techniques. The study opens the way to explore cellular processes at atomic resolution to better understand molecular mechanisms of life, and to use this knowledge to improve our ability to treat diseases.

Recently, in our previous paper, the ``coin-turning walk'' was introduced on ℤ. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in ℤ<sup>d</sup>, d≥2: at time n the direction of the process is ``updated'' with probability p_n; otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits.

Our results complement previous ones in the literature about ``correlated'' (or ``Newtonian'') and ``persistent'' random walks.

This is joint work with Stas Volkov (Lund, Sweden).

Random variables in Voiculescu’s free probability theory can model the limits of suitable random matrix models. The Brown measure of a free random variable is a replacement for the eigenvalue counting measure of square matrices. The circular operator is a natural limit operator for non-normal random matrices with i.i.d. entries, in the sense that these random matrices converge in moments to the circular operator. The circular law says that the empirical eigenvalue distribution of the i.i.d. random matrix converges to the Brown measure of the circular operator, which is the uniform measure in the unit disk.

Let $c$ be a circular operator, and $g$ be an elliptic deformation of $c$, and $x$ be a free random variable. We obtained a Brown measure of $x+c$. The Brown measure of $x+g$ is not calculated directly but is shown to be the pushforward measure of the Brown measure of $x+c$ under some explicitly constructed self-maps of the complex plane. There Brown measures are related to the limit empirical eigenvalue distributions of deformed i.i.d., deformed GUE, and deformed elliptic random matrices.

We consider the random field h(t, x) called KPZ fixed point. We allow it to live on the line and on the ring. We assume the narrow wedge initial condition for it. The corresponding multipoint distribution has representation formula as contour integral of the Fredholm determinant of integrable operator. We establish integral formula for this Fredholm determinant in terms of solutions of matrix KP equation. We did not find initial conditions for these solutions. In the case of one point distribution they are singular and solution can’t be constructed using standard inverse scattering methods. We identify differential equations for Fredholm determinant mentioned above as multicomponent KP equations, generalizations of Tracy-Widom equations for the multipoint distribution of the Airy process, generalization of the Adler-Van Moerbeke equation for the two-point distribution of the Airy process. We can also notice that the equations mentioned above hold for the large class of integrable operators, which we called cubic integrable operators. This talk is based on the joint work with Jinho Baik and Guilherme Silva (arXiv:2208.11638).