13 problems

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

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…

Brownian-bridge failure-tail conjecture. The probability that the failure size in the giant component exceeds $\kappa(k)$ behaves like the probability that this Brownian bridge sta…

Let $\mathcal L$ be the directed landscape, let $\pi^*(\tau T)$ be the geodesic location at time $\tau T$, and condition on $\mathcal L(0,0;0,T)=L$. For $\tau\in(0,1)$, consider th…

Let ${\mathsf H}_{\rm step}$ be the KPZ fixed point with step initial condition, and let ${\mathsf H}_{{\rm step},L}$ denote its conditional version under…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $b$ is a standard Brownian bridge whose average is represented by…

Let $n$ be a non-negative integer. For independent standard Brownian bridges $b_0,\ldots,b_{2n}$, define … Here $\max$ denotes the maximum over the bridge parameter. Let…

Let $n$ be a non-negative integer and let $h_0$ be an arbitrary sufficiently regular continuous function satisfying … For independent standard Brownian bridges $b_1,\ldots,b_n$, de…

Conditioned renewal-process FCLT conjecture. As $n\to\infty$, the appropriately scaled conditioned renewal process should satisfy a counterpart to the functional central limit theo…

Let $V(\alpha)$ denote the optimal stopping value function for the $\alpha$-Brownian bridge, as defined in the paper. Numerical results indicate that $V(\alpha)$ has a local minimu…

Chafaï's conjecture. The sequence $\mathcal W^{(n)}$ converges in distribution to the tied-down Brownian bridge, with the same normalization as in Theorem 1:…