21 problems

Least-likely-state conjecture. In the stationary distribution, the least likely states are precisely those that are reversed-speed ordered.

Let $\Lambda$ be a finite graph, and let $L$ be the generator of the symmetric simple exclusion process with positive jump rates $J_{xy}$ on its edges. Let $H_\Lambda^{(n)}$ be the…

Universal backward-geodesic endpoint conjecture. There exists a universal constant $c_2$ such that

Universal Airy-one scaling conjecture. There exist universal constants $c_1$ and $c_2$ such that

Let $D_N(t)$ be the discrepancy between the backward-geodesic endpoint index $N(t\downarrow0)$ and the corresponding minimization index in the exclusion-process setting considered…

ASEPsc particle-position conjecture. For any $N\in\mathbb{Z}$ and every $s\in\mathbb{R}$,

ASEP backward-geodesic endpoint conjecture. For any $N\in\mathbb{Z}$,

Let an impurity in the totally asymmetric simple exclusion process have jumping rates $b1$ and $b2$, and initially be located at the interface of a $1$-$0$ profile. Assume that…

Consider the symmetric simple exclusion process with reservoirs on a segment of size $N$, with parameters $p$ and $q$, and let $t^*$ denote the characteristic mixing-time scale. In…

Mixing-time and cutoff conjecture. The following claims should hold:

Jara's conjecture. The mixing time is of order

Lacoin's conjecture. The mixing time is of order

Steady-state ratio conjecture. The following statements are equivalent: $t=1$; and, for all such partitions $\lambda$ and all permutations $\mu,\nu\in S_n(\lambda)$,

Consider the simple exclusion process with open boundaries, with phase parameters $a$ and $b$, and let $t^N_{\mathrm{mix}}(\varepsilon)$ denote its $\varepsilon$-mixing time. Maxim…

Consider the asymmetric simple exclusion process with open boundaries, with parameters $a,b,p$ satisfying the assumptions of the paper's two-sided asymmetric regime. For…

Symmetric cutoff conjecture. The lower bound in the symmetric pre-cutoff estimate is sharp, and cutoff occurs.

Let a Galton–Watson tree be supercritical and condition it on survival. Consider the simple exclusion process on this tree in the constant speed model and in the variable speed mod…

Let $b_N$ and $k_N$ satisfy … for some $\beta\in(0,\infty)$, with $\lim_{N\to\infty}k_N/N=0$. Write $T_{\rm mix}^{N,k_N}(\varepsilon)$ for the mixing time of the weakly asymmetric…

Let $M_{i,i+1}^{(\ell)}$ be the multi-state ASEP update matrix formed from the fused generators $M_{i,i+1}^{(\ell;r)}$. TASEP-limit conjecture. The reparametrization … gives a prop…

Let $M_{i,i+1}^{(\ell)}$ be the update matrix … Normalize b_r^{(\ell)}=\begin{bmatrix}\ell\r\end{bmatrix}\widetilde b_r^{(\ell)}, where … Because the update matrix is symmetric…

Let the inhomogeneous multispecies exclusion process have $n$ classes of particles, with particles of class $i$ jumping at rate $x_i$. Let $w(\boldsymbol{\rho})$ denote the station…