47 problems

Orthogonality conjecture. In the limit $c\to\infty$,

KPZ universality conjecture. All KPZ models, under the so-called $1:2:3$ scaling, converge to the KPZ fixed point.

Particle-conjugation symmetry conjecture. The following equality holds:

Let $K_1$ be the product of the first three matrices in the displayed factorization of the TASEP kernel, and let $K_2$ be the operator defined by … Here the operators act on the in…

Consider the constant-$q$ specialization in which the limit shape reaches the corner; in particular, the paper notes the case $b=-\frac{1}{c}$ for which the limit shape for $q>1$ e…

Consider the generalized principal specialization $x_i=q^{i-1}$ and $y_j=q^{b(j-1)}$, in the regime where the resulting limit shape is linear rather than flat at the corner. Let th…

Tableau-process weight conjecture. This weight is proportional to the weight derived from the tableau process of Ayyer et al.

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 open ASEP with boundary parameters. In the three-parameter case, the basic weight function is described using bi-colored Motzkin paths, while general boundary paramete…

Let $u,v\in\mathbb R$. Let $\mathbb{P}_{\operatorname{KPZ}}^{u,v}$ denote the measures on paths constructed from the process $B_1=W+X$, where $W=(B_1+B_2)/2$ and $X=(B_1-B_2)/2$, w…

In the setting of the section, consider right Bernoulli TASEP with sequential updates and an additional caterpillar of length $b\varepsilon^{-1/2}$ placed at distance…

Consider discrete-time right Bernoulli TASEP with sequential updates and an additional caterpillar of length $M=b\varepsilon^{-1/2}+c$ placed at distance $M'\geq M-1$ from the righ…

Let $q\in(0,1)$ and let $n$ be the parameter determining the one-species jump rates. Define … Consider the type $D$ ASEP with parameters $(2,0)$ and $(3,0)$ on the infinite lattice…

Fix $r_1,\dots,r_n\in\mathbb{Z}$, $x_1,\dots,x_n\in\mathbb{R}$, $t>0$, and an initial height function $h$. Let $K_r$ be the kernel defined using these parameters, and let…

Let a model in the KPZ class have a height function $h$ and initial data $h_0$. Suppose that … Then there are non-universal constants $c$ and $C_\varepsilon$ such that … where…

In the half-space beta polymer, vertices $(x,x)$ on the boundary have three associated edges, and the parameter $t$ currently restricts the boundary weights so that one edge has ze…

Let $T_0,\dots,T_{n-1}$ be the absorption times in the type $B$ oriented swap process, and let $L((0,0),(u,v))$ denote the corresponding half-space LPP passage time. Type B distrib…

The type $B$ oriented swap process has absorption times $T_0,\dots,T_{n-1}$, and the half-space TASEP admits a shift-invariance identity analogous to the type $A$ identity. Type B…

Let $\alpha>0$, $u>-\alpha$, and let $z_{u,v}$ be the explicitly constructed stationary processes indexed by $v\in(-\alpha,\min(u,0)]$ for the half-space log-gamma polymer. For ini…

Let $E$ be the underlying space, let $\mathbb{P}$ be a point process on $E$, and let $\rho^R$ be an $L^1$-specialization. The pair $(\mathbb{P},\rho^R)$ is Giambelli compatible whe…

The paper studies wave functions from the Calogero–Moser/Ruijsenaars–Schneider family in QQ-dual form, represented as sums over colored paths in a discrete stochastic model. Genera…

The paper considers a quantum mechanical system represented through an auxiliary statistical stochastic model. A wave function $\Psi_E(x)$ admits a Feynman representation as a sum…

Let $F(x;\tau,\gamma)$ denote the one-point distribution for periodic TASEP, let $F_{\mathrm{GUE}}$ denote the GUE Tracy–Widom distribution, and let $x\in\mathbb R$. The parameters…

Spin Whittaker orthogonality conjecture. For all $\underline{L}_N,\underline{L}'_Ninmathpzc{W}_N$,

Consider the space of self-duality functions for the asymmetric particle systems in question, and restrict attention to self-duality functions having a nested product form. In the…