6 problems

Recurrence conjecture. The model is recurrent in $d=2$ even with weak reinforcement.

Construct a random hierarchical lattice by starting from two seed graphs $G$ and $H$, choosing independently at each edge-replacement step either $G$ with probability $q$ or $H$ wi…

Let $p_c$ be the critical threshold of a hierarchical lattice. Distinguish the lattice's local structure from its global characteristics, such as its macroscopic dimension. Localit…

Let $R_n$ denote the effective resistance between $0$ and $1$ at level $n$ in the unit-resistance setting. Hambly–Kumagai's resistance-exponent conjecture. There exists $\alpha$ su…

Let the Dyson hierarchical lattice have critical dimension $2$, and let $\overline{W}$ be any inverse temperature. Consider the vertex-reinforced jump process (VRJP) on this lattic…

Let $b\in \\{2,3,4,\ldots\\}$ and assume $s=b$. For $r\in\mathbb R$, let $\beta_{n,r}^{(b)}$ be the inverse-temperature sequence from Theorem, and let $W_n(\beta)$ denote the norma…