7 problems

Let $(X_n)_{n\geq 0}$ be a random walk in a stationary ergodic environment on $\mathbb{Z}^d$. A deterministic limiting velocity is a vector $v$ such that … for almost every realisa…

Let $(Z_n)_{n\geq 0}$ be the random walk defined in Section 2, and let $\mathbb{P}$ denote its probability measure. Law of large numbers conjecture. There exists…

Let $\mathsf{G}$ be a finite graph on $\mathsf{N}$ vertices, let $D$ be its adjacency matrix, and let $\mathsf{T}_i$ be a periodic tree directed cover of $\mathsf{G}$ with root of…

Macdonald law-of-large-numbers conjecture. With almost sure convergence,

Positive-Plancherel-parameter conjecture. The technical assumption $\gamma=0$ in the law of large numbers can be dropped: the same convergence should hold for every triplet…

Fix a dominant integral weight $\Lambda$ and a sequence $(w^{(k)})$ in $W^{\mathrm{aff}}$ such that $l(w^{(k)}) \to \infty$. Let…

Durrett–Rogers law of large numbers conjecture. Under these assumptions, $X(t)/t\to0$ almost surely. This conjecture concerns the long-time displacement of a one-dimensional self-i…