12 problems

Let $X$ be a $p_n$-ERW in direction $ellinmathbb{S}^{d-1}$ on $\mathbb{Z}^d$, where $dgeq 2$ and $p_n=\mathcal{C}n^{-beta}wedge 1$ with $betageq 1/2$. Let $\mathcal{R}_n^X$ be the…

Let $X$) be a $p_n$-ERW in direction $ell in \mathbb{S}^{d-1}$ on $\mathbb{Z}^d$, where $dgeq 3$ and $p_n=\mathcal{C}n^{-1/2}wedge 1$. Let $\widehat{B}^n$ denote the associated res…

Decay-exponent conjecture. The positive-probability transience result should hold for every

Transience and growth conjecture. The walk should satisfy

Let $\boldsymbol{Z}=(Z_n)_{n\ge 0}$ be the balanced excited random walk on $\mathbb Z^2$: on the first departure from each site, it takes a vertical step to one of the two neighbor…

Let $\mathbf{X}$ be the $(L,1)$ non-nearest-neighbour excited random walk described above, and let … be its expected total drift. Davis–Peterson conjecture. The limiting speed of t…

Volkov's conjecture. Any cookie random walk which moves, after excitation, like a simple random walk is recurrent.

Let $\mathcal{T}$ be a tree containing the binary tree, and consider the $M$-digging random walk: each vertex has $M$ cookies, and whenever the walk visits a vertex with a cookie r…

Let $X_n$ be an excited random walk in a cookie environment with Markovian cookie stacks, and let $\theta$ and $\tilde\theta$ denote the associated parameters. A…

Velocity conjecture. There exists a deterministic constant $v_0\in[0,\infty)$ such that

Extension conjecture. Results for nearest-neighbor excited random walks that are characterized by the parameter $\delta$ should extend, under appropriate assumptions, to excited ra…

Ergodic-environment conjecture. Theorem should extend to ergodic cookie environments, possibly under some additional ellipticity assumptions; that is,