5 problems

In a one-dimensional diffusive system, consider fluctuations of the time-integrated current over a long observation time and the optimal trajectories governing their probability. A…

The extension conjecture. The displayed formula, with the stated contour prescription and singlet initial vector, holds for every $n\geq 3$ as well.

Maximal-current current-variance conjecture. Lemma 1 extends to the entire maximal current phase of the open ASEP.

Spohn's conjecture. The equilibrium fluctuations of the current should converge in the limit to a $1/4$ fractional Brownian motion.

The partially asymmetric simple exclusion process is described using a deformed Markov matrix, with deformation parameter $\xi$ and system variables $\mathbf{x}$. For each positive…