7 problems

Consider a finite transitive graph with $n$ vertices, coalescing random walks, coalescence time $\tau_{\mathrm{coal}}$, relaxation time $t_{\mathrm{rel}}$, and hitting time scale…

Let the directed spanning forest (DSF) be defined in Euclidean space with dimension $d$ and exponent $p$. Its trajectories are the directed paths obtained by iteratively following…

Fractional Hausdorff dimension conjecture. In a suitable topology, the limiting set has fractional Hausdorff dimension

Aldous–Fill conjecture. Under some mixing conditions on the random walk on ${\mathbb G}_N$, the expected coalescence time $E[C_N]$ should be of the same order as $h_N$, as in the m…

Let $G=(V,E)$ be any graph, let $u\in V$, and let $(x_0,x_1,\ldots,x_{t_{\operatorname{hit}}})$ be a path of vertices, meaning that for every $i$, either $x_i=x_{i+1}$ or…

Fix $d\in\mathbb{N}$ and let $G$ be a $d$-regular graph. Let $p$ be the parameter of the constrained Ising process on $G$, let $\tau_{\mathrm{mix}}^{\mathrm{RW}}$ denote the mixing…

Let $G$ be a connected graph with $n$ vertices. Let $C(n)$ denote the expected coalescence time of $n$ coalescing random walks, initially one at each vertex, and let…