The critical two-dimensional stochastic heat flow (2d SHF) is a two-parameter process of random Borel measures on R^4 derived in a breakthrough article by Caravenna, Sun, and Zygouras as a universal distributional limit of point-to-point partition functions for (1+2)-dimensional models of a directed polymer in a random environment within a critical weak-coupling scaling regime. I will discuss continuum polymer measures associated with the 2d SHF, with an emphasis on the structure of their second moments. Our approach is inspired by a one-dimensional continuum directed polymer model formulated by Alberts, Khanin, and Quastel.

In this talk, I will present the transition probability of TASEP in half-space starting with general deterministic initial condition. I will show how properties of half-space TASEP are manifested on the formula. Taking the 1:2:3 KPZ scaling, I will present the transition probability of the half-space KPZ fixed point.

We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. We then establish optimal fluctuation exponents for the height function in both the subcritical and critical regimes, along with corresponding estimates for the polymer endpoint. Based on a joint work with Yu Gu.

This talk will discuss some recent progress on understanding the structure of semi-infinite geodesics and their associated Busemann functions in the inhomogeneous exactly solvable exponential last-passage percolation model. In contrast to the homogeneous model, this generalization admits linear segments of the limit shape and an associated richer structure of semi-infinite geodesic behaviors. Depending on certain choices of the inhomogeneity parameters, we show that the model exhibits new behaviors of semi-infinite geodesics, which include wandering semi-infinite geodesics with no asymptotic direction, isolated asymptotic directions of semi-infinite geodesics, and non-trivial intervals of directions with no semi-infinite geodesics. Based on joint work with Elnur Emrah (Bristol) and Timo Seppäläinen (Madison)

We implement a version of conformal field theory (CFT) that gives a connection to SLE in a multiply connected domain. Our approach is based on the Gaussian free field and applies to CFTs with central charge c=1. In this framework we introduce the generalized Eguchi-Ooguri equations and use them to derive the explicit form of Ward's equations, which describe the insertion of a stress tensor in terms of Lie derivatives and differential operators depending on the Teichmuller modular parameters. Furthermore, by implementing the BPZ equations, we provide a conformal field theoretic realization of an SLE in a multiply connected domain, which in particular suggests its drift function, and construct a class of martingale observables for this SLE process.

TBA