The five-site localization conjecture for vertex-reinforced random walks

Let $w$ be a nondecreasing weight sequence satisfying

$$\sum_{n=0}^{\infty}\frac{1}{w(n)}=\infty.$$

Extend $w$ to the positive reals by $w(t)=w(\lfloor t\rfloor)$, define

$$W(t)=\int_0^t\frac{1}{w(u)}\,du,$$

and let $W^{-1}$ be its inverse. Set

$$I_\alpha(w)=\int_0^\infty\frac{dx}{w\bigl(W^{-1}(W(x)+\alpha)\bigr)},$$ $$\alpha_c(w)=\inf\{\alpha\geq 0:I_\alpha(w)<\infty\},$$

and, with $W^{-1}(u)=0$ for $u<0$,

$$J_\beta(w)=\int_0^\infty\frac{dx}{w\bigl(W^{-1}(2W(x)+\beta)\bigr)},\qquad \beta_c(w)=\inf\{\beta\in\mathbb{R}:J_\beta(w)<\infty\}.$$

Let $R'$ be the set of sites visited infinitely often. Five-site localization conjecture. If $\alpha_c(w)=\infty$, then

$$\mathbb{P}(|R'|=5)>0\quad\Longleftrightarrow\quad \mathbb{P}(|R'|=5)=1\quad\Longleftrightarrow\quad \beta_c(w)<\infty.$$

This conjecture characterizes both positive-probability and almost-sure localization on five sites. The almost-sure part is described as particularly difficult; the paper recalls that it is known for linear weights, while the general characterization remains open.