In random matrix theory the Tracy-Widom GOE distribution describes the location of the largest eigenvalue of large real symmetric random matrices. It also describes height fluctuations of certain random growth models in the KPZ universality class, and can be characterized as the maximum value of the Airy process minus a parabola. This characterization has a natural deformation when the Airy process is replaced with the Airy process with wanderers, a process whose marginal distributions correspond to spiked complex Hermitian random matrix models like GUE.

I will characterize the distribution of this maximal height in several ways: with a Fredholm determinant formula; in terms of Painlevé transcendents; as a KPZ fixed point distribution with a particular initial condition; and as a solution to two PDEs, the KdV equation and the Bloemendal-Virag equation for spiked random matrices. This is joint work with Daniel Remenik and Gia Bao Nguyen.

In this talk, we will discuss the geodesic of the directed last passage percolation (DLPP) with iid exponential weights. We will show how to find the explicit one-point distribution and its limit in the KPZ scaling. This is the first formula of the one-point distribution of the point-to-point geodesics in the KPZ university class, and is believed to be universal for the models in this class. Our method relies on a transition probability formula of the DLPP obtained by Johansson, some summation techniques developed by Baik and Liu, and a Cauchy-type summation identity found by Liu.

Conformational transitions are important in biochemical reactions, which are usually described by irreversible drift-diffusion processes. We will first summarize several reversibility criteria. Then we propose two upwind finite volume schemes for the Fokker-Planck equation with or without the information of the invariant measure. Both two schemes enjoy stochastic Q-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, which enable us to prove unconditional stability, ergodicity and error estimates. With the optimal control reinterpretation for transition path theory, we further give an optimally controlled random walk on collected point clouds which enables efficient Monte Carlo simulations for mean transition path in biochemical reactions.

The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a close relative of the discrete Gaussian free field (DGFF), for which instead the most likely interfaces are those preserving the mean height. However working with the two models presents some key differences. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will investigate a random walk representation for the covariances of the MM and by means of it we will define and study the MM on trees. This talk is based on joint work with Biltu Dan (IISc Bangalore), Rajat Subhra Hazra (University of Leiden) and Rounak Ray (TU/e).

The fascinating diversity of material behavior at the macroscopic scale, including plasticity, phase transformations, viscoelasticity or diffusion, can only emerge from the underlying atomistic or particle behavior. Yet, the direct connection between these two scales for non-equilibrium phenomena remains an extremely challenging quest from both a theoretical and computational perspective. This knowledge gap currently hinders predictive simulations and material discovery, and leads to large economic losses and barriers to innovation in many industrial sectors. In this talk, we will discuss several advances in the predictive modeling of non-equilibrium mechanics, by combining tools from continuum mechanics, statistical physics, applied mathematics, as well as machine learning and data-driven strategies.