12 problems

Equality of the tail constants. The constants $C_1$ and $C_2$ are equal. This would sharpen the established two-sided asymptotic-order estimate for the tail probability of the whol…

Let $X$ be a random walk in Dirichlet environment on ${\mathbb Z}^2$ with positive parameters $(\alpha_1,\ldots,\alpha_4)$. Define … A local accelerating function $\gamma$ is a pos…

Let $K_n$ be the complete graph on $n$ vertices, let $\omega$ be the edge-disorder environment, and let ${{\mathcal T}}^\omega_{n,\beta_n}$ be the random spanning tree in random en…

Consider the long-range contact process (LoRaC) on the spatial model of the paper, with edge length $|y-x|$ and a parameter $\alpha>0$. Fewer edges. The LoRaC has a phase transitio…

Let $q^{(t)},p\in(0,1)$ be given and let $\alpha>1$. Let $q^{(x)}$ denote the parameter controlling the sparse spatial environment in the long-range contact process (LoRaC). Extinc…

Let $q^{(t)},q^{(x)},p\in(0,1)$ be given, and let $\alpha>1$. The long-range contact process (LoRaC) is the percolation model described in the paper, with edges whose length distri…

Let $N$ denote the size of the segment, and consider the simple exclusion process in a ballistic random environment. Its mixing time is the time needed for the process to approach…

Let $z=(z(n))_{n\geq 0}$ be a path in the admissible path space, let $z^k=z(1),\ldots,z(k-1)$ denote its past, and let $\widehat z_m(k)=z(m+k)-z(m)$ with…

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 $(X_n)_{n\geq 0}$ be the random walk in a random environment on a tree, let $|X_n|$ denote the distance of the walk from the root, and let $\mathbb{P}$ be the governing probabi…

Let $(X_n)_{n\geq 0}$ be the random walk in a random environment on a tree, let $V(X_n)$ denote its potential energy, and let $\mathbb{P}$ be the governing probability measure. Ass…

Let $\lambda_p^{u_0}(\kappa)$ denote the quenched Lyapunov exponent of order $p\in\mathbb{N}$ for mobility parameter $\kappa$, and let $\lambda_0^{u_0}(\kappa)$ be the correspondin…