5 problems

Consider scale-free percolation in the polylogarithmic regime, with parameters $\tau>3$ and $\alpha\in(1,2)$, and let $\Delta(\alpha)$ denote the exponent governing graph distances…

Consider scale-free percolation (SFP), the geometric inhomogeneous random graph (GIRG), and the hyperbolic random graph (HRG), and let $\mathcal{B}(x,k)$ denote the graph-distance…

Let $\gamma_{0}^\star=(0,v_1^\star,v_2^\star,\dots):=\arg\min\{\lvert\gamma_L\rvert\}$ be the shortest exploding path starting at $0$, where $\lvert\gamma_L\rvert$ denotes its $L$-…

Consider a geometric inhomogeneous random graph (GIRG) on a box of volume $n$, with vertex weights following a power-law distribution of exponent $\widetilde\tau\in(2,3)$. Add i.i.…

Let $d_L(x,y)$ denote the $L$-distance between vertices $x,y\in\mathbb{Z}^d$, and let $V_x$ be the explosion time from $x$. Under the condition of the distance theorem, these quant…