4 problems

Let $G$ be a transitive transient graph. For $\varepsilon>0$, let an $\varepsilon$-$\operatorname{WUSF}$ be obtained by independently retaining each edge of a wired uniform spannin…

Let $X$ be the increment random variable of the random walk in $\mathbb{R}^d$, and let $G_n$ denote its centre of mass after $n$ steps. Assume that the support of $X$ is not contai…

Let $P$ be a distribution on environments satisfying Assumption asmiid, and suppose that … Here, $\mathfrak{R}$ denotes the first return time to the origin, and $X_1$ is the first…

Let $\mu_1,\ldots,\mu_{d-1}$ be $d$-dimensional measures in $\mathbb{R}^d$, with $d\geq 4$, zero mean, and finite $(2+\beta)$-moments for some $\beta>0$, and let $\ell$ be an arbit…