8 problems

Consider the two-walker model in which the walker to move is selected randomly, and let $M^{\rm rd}_\infty$ denote the limiting proportion of the left edge weight. The alternating-…

Biased edge-reinforced random walk transience conjecture. The $\lambda^\ast$-biased edge-reinforced random walk is transient whenever $\lambda\neq1$.

Alternating two-player urn distribution conjecture. The random variable $Y\in[0,1]$ has a density with respect to Lebesgue measure on $[0,1]$, is not Beta-distributed, and has a di…

Dichotomy conjecture. One has

Let $d$ be fixed, and consider the once-reinforced walk on a $d$-regular tree with forward-edge weights $u_0$ before reinforcement and $u_1$ after reinforcement. Monotonicity conje…

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…

Let $\alpha>1/3$, define $\alpha_1:=+\infty$ and, for $L\ge2$, … For the walk, let $R'$ be the set of sites visited infinitely often, … Erschler–Tóth–Werner's conjecture. If…

Let $D$ be the state space of pairs of occupation measures, let $x(n),y(n)$ denote the two occupation-measure processes, let $z(n)=(x(n),y(n))$, and let $U$ denote the uniform occu…