10 problems

Let $q\in[0,1)$ be the jumping rate, let $U_0$ denote the speed of particle $0$, and write $x=\tfrac{1+u}{2}$ when conditioning on $U_0=u$. Let $\mathcal{C}_0^{n,(q)}$ be the set o…

Let $u$ solve the hydrodynamic equation … with boundary parameters determined by the reservoir rates $(\alpha,\beta,\gamma,\delta)(t)$ through the relations defining $\nu_\pm(t)$.…

The half-line KPZ stationary measures described above are considered in the large-scale limit as $x$ tends to infinity; this limit is expected to define stationary measures of the…

GOE Tracy–Widom cutoff-profile conjecture. The cutoff profile for the multi-species ASEP should be the GOE Tracy–Widom distribution function, and the cutoff window should have orde…

KPZ universality conjecture. The KPZ class is conjectured to be governed by universal scaling exponents $-1/3$ for fluctuations and $2/3$ for correlations, together with limit laws…

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…

Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_m)$ be a partition, and let $K_{\lambda}(\xi)$ denote the Koornwinder moment defined by … where $Z_N$ is the ASEP partition function…

Consider the ASEP height function with flat initial data, meaning that initially the particles occupy the even positive sites. The long-time behavior is studied after centering and…

Completeness conjecture. The eigenvalues obtained from these equations with $n$ pseudo-excitations, together with the eigenvalues given by the corresponding equations in the comple…

Let $p$ and $q$ be the ASEP jump parameters with $p<q$, let $x_m(t)$ denote the position of the $m$-th particle at time $t$, and let $K_0$ be the operator on $L^2(\mathbb {R})$ wit…