5 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 vertex-reinforced non-backtracking random walk on a connected finite undirected graph, with reinforcement function $W(k)=(1+k)^\alpha$. Let $C=\{i_1,\dots,i_\ell\}$ be…

Condition on a vertex-reinforced random walk localizing on a finite subset. Let $x$ denote a stable equilibrium of the associated differential equation, let $S(x)$ be its support,…

Let $(X_n)_{n\in\mathbb{N}}$ be a vertex-reinforced random walk on $\mathbb{Z}$ with polynomial reinforcement $W(n)=n^\rho$, where $\rho<1$. In this regime, the walk almost surely…