54 problems

Consider the non-conservative symmetric simple exclusion process on a segment of size $N$ with reservoirs, and let $t_{\mathrm{mix}}(\epsilon)$ denote its mixing time for fixed…

The mixing time is the time required for a Markov particle system to approach its stationary distribution. A particle system is in the maximal current phase when its stationary par…

The $t^{1/3}$ superdiffusivity conjecture. In particular, there is a constant $C$ such that

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…

Universality conjecture. Throughout phases A and B,

Let the symmetric Dyson exclusion process (SDEP) be the lattice gas with exclusion and long-range Coulomb-type interactions described in the paper. At macroscopic scales, the SDEP…

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 $s\in\mathbb{R}$,

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

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…

Product-Bernoulli determination conjecture. For every $\alpha\in[1,3/2]$, the renormalization constant $\mathscr{R}_{N}$ is still determined by product Bernoulli measures.

Speed-selection conjecture. Under mild conditions, if $Q$ is transient or null recurrent, the leftmost particle still has speed $v_0$; if $Q$ is positive recurrent, the speed $v$ s…

Hydrodynamic-limit conjecture. The main result should remain valid for a fairly general class of initial distributions, beyond the supercritical ergodic configurations treated in t…

The Type D asymmetric simple exclusion process is a two-species interacting particle system on a one-dimensional lattice, with parameters $(q,n,b4)$, where $q$ is the asymmetry par…

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…

Cutoff without the reservoir condition. Cutoff should still occur without being restricted by the condition $\frac{1}{1-p}=N^{o(1)}$. In particular, the conjecture asserts cutoff a…

Consider a finite inhomogeneous particle system with exclusion interaction, where $N+1$ particles have constant drifts satisfying $b_i-a_i\u2261 u$ for all $i\in[N+1]$. For $N=3$,…

Jara's conjecture. The mixing time is of order

Lacoin's conjecture. The mixing time is of order

High-density ETIS conjecture. At high density, every ETIS state belongs to one of the families described for the F-SSEP; equivalently, there are no other ETIS states.

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