77 problems

Glynn–Whitt constant conjecture. Based on simulations, they conjectured that $c=2$. This conjecture concerns the explicit value of the constant in the Brownian-percolation asymptot…

Consider the continuous last-passage percolation model in $[0,1]^d$, with an optimal path given by the closure of … For $d\geq 3$, jumps in this path may cross lower-dimensional se…

Let $A^*$ be an optimal path in the continuous model, let $Y_i$ denote its random locations, and let $P^*$ be the connected increasing extension constructed from the closure of the…

Let $A_{n,k}$ be an $n\times k$ random matrix with independent and identically distributed entries, and let … Here $Y_{n,k}$ is a sample covariance-type matrix, and its coefficient…

Invariant measures have been constructed for several half-space models, including geometric and exponential last-passage percolation, the inverse-gamma polymer, and the KPZ equatio…

For half-space geometric last-passage percolation, Dauvergne and Zhang constructed a family of jointly invariant measures, meaning invariant measures describing several functions s…

Half-space geometric last-passage percolation has invariant measures, and a family of such measures was constructed by Barraquand and Corwin. An extremal invariant measure means an…

A half-space Gibbsian line ensemble is expected to have a scaling limit called the pinned half-space Airy line ensemble. Locally, the curves are ordered line ensembles with pinning…

Let $\tilde{\beta}$, $\tilde{t}$, $\alpha$, and $A$ be the parameters from the cited high-density asymptotic theorem, and let $\widetilde{Q}_{\tilde{\beta}<\tilde{t}}^{High}$ be th…

Consider the dynamical site-resampling setting for last-passage percolation, and let $E$ be the set of times at which non-trivial bigeodesics exist. A bigeodesic is a bi-infinite d…

Consider planar last-passage percolation with geometric weights. Let $T_n(\omega)$ and $T_n(\tilde\omega^t)$ denote the last-passage times in the original and site-resampled enviro…

The directed landscape $\mathcal L$ is a four-parameter scaling-limit object constructed from Brownian last passage percolation. Universality conjecture. It is conjectured to be th…

Let $L_{m,n}^{\bm\alpha,\bm\beta}$ denote the exponential last passage percolation time with thick-boundary parameters, and let $x_i$ and $y_j$ be the associated boundary parameter…

Conditional geodesic fluctuation conjecture. Conditional on $\mathcal L(aN,bN)=\ell N$,

Variational conditional law of large numbers conjecture. For $(x,y)\in\mathbb R_+^2$,

High-time-spine conjecture. The geodesic to $(xaN,ybN)$ follows the high-time spine from $(1,1)$ to $(taN,tbN)$ for some $t\in[0,1]$, and then grows typically from $(taN,tbN)$ to…

Conditional fluctuation conjecture. The following hold. For $(x,y)\in\Omega_1$, the centered passage time has Tracy–Widom $\operatorname{TW}_2$ fluctuations on the $N^{1/3}$ scale…

Brownian-bridge independence conjecture. The Brownian bridges governing the fluctuations in the below-diagonal and above-diagonal regimes are independent.

Independence conjecture. Conditional on the one-point upper large-deviation event, the limiting distributions associated with points below the diagonal and points above the diagona…

Conditional law of large numbers conjecture. The convergence in the conditional law of large numbers holds with

Let $\frac12\log q<0$, with $[?]$?

A bi-infinite geodesic is a geodesic path extending infinitely in both directions in a model of the KPZ universality class. Nonexistence conjecture. With probability one, bi-infini…

Uniform-path comparison conjecture. For all $t e(0,1/2]$,

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

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