4 problems

Endpoint conjecture. For every weight $w$,

Let $w$ be a nondecreasing weight sequence satisfying … Extend $w$ to the positive reals by $w(t)=w(\lfloor t\rfloor)$, define … and let $W^{-1}$ be its inverse. Set … … and, with…

Let $X$ be a one-dimensional vertex-reinforced random walk with reinforcement weights satisfying $w(n)\sim n^\alpha$. The walk is recurrent if it visits every site infinitely often…

Consider a vertex-reinforced random walk on the integer lattice $\mathbb{Z}$ with linear weight $w(n)=cn+1$, where $c>0$. The walk localizes on $5$ sites if it visits only those…