Week 26.01.2026 – 01.02.2026

The Critical 2d Stochastic Heat Flow arises as a non-trivial solution to the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point. It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos. We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF and outline some of its features and related questions. This will be mostly based on joint works with Francesco Caravenna and Rongfeng Sun but contributions from other researchers in this endeavour will also be mentioned.

Periodic PushASEP model is a bidirection interacting particle system with N particles moving on a torus of size L. To solve the system, we apply the Bethe Ansatz to compute the Fourier transform of the joint Markov process (X, Q) with respect to Q, where X is an N-tuple denoting the particle positions, and Q is the total current of the system. In particular, this can be written as a (N+1)-fold contour integral, which, by residue computations, simplifies into a 1-fold contour integral, then to a sum over Bethe roots. This leads to the two following applications: 1) The 1-fold contour integral is ready for asymptotic analysis, that we can perform at the relaxation time scale, giving similar limiting distribution as in [Baik and Liu, 2018]. 2) The sum over Bethe roots corresponds to the spectral decomposition of the system evolution, and when time t=0, it justifies rigorously the completeness of the Bethe eigenfunctions. Based on joint work with Axel Saenz (Oregon).