11 problems

Let $N$ be a positive integer and let ${\theta}(k:l)$, for $k,l\geq 0$, be the parameters defining the generator ${\mathcal A}_N^\theta$; the boundary values ${\theta}(k:0)$ and…

Let the $cell_2$ Euclidean directed spanning forest (DSF) be viewed under diffusive scaling, and let the resulting DSF network be the collection of its diffusively rescaled traject…

Brownian-web convergence conjecture. In a diffusive limit, the trajectories of the process $ceta(t)$ converge to the Brownian web in the spacetimes-time band $[0,1]\times\mathbb R$…

Let $D^{\mathrm{Br}}$ denote the Brownian web distance, let $\mathfrak A(y)=\mathcal L(0,0;1,y)$ be the parabolic Airy process associated with the directed landscape $\mathcal L$,…

The Brownian web distance is a directed metric on the tree formed by coalescing simple random walks, and its scale-invariant limit is the Brownian web distance. The full directed l…

Baccelli–Bordenave conjecture. The scaling limit of the DSF on the Poisson points is the Brownian web $\mathcal W$.

Let $\Gamma_s^\epsilon$ be the set of right-most infinite open paths at dynamical time $s$. Let $S_{\alpha(p),\sigma(p),\epsilon}$ denote the corresponding space-time rescaling, an…

Let $N$ be the backbone under consideration, and call a compact region enclosed by the left-most and right-most paths that start at a separation point and meet for the first time a…

Fix $p\text{ in }(p_c,1)$, let $\Gamma$ be the collection of all rightmost infinite open paths in the supercritical oriented percolation configuration, and let $\alpha=\alpha(p)>0$…

Let $\pi^0\in{\cal W}_0$ be a path and let $\rho_t(\pi^0(t))$ denote the mass associated with its position at time $t$. Mass-variation conjecture. The function … has locally unboun…

Let a directed navigation on a random set of points in the plane produce an associated Yao graph or $\Theta$ graph, and let the construction studied by Fontes, Isopi, Newman and Ra…