15 problems

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying fa…

Let $m(0)$ denote the odometer at the origin for fixed-energy activated random walk with i.i.d. initial particle density $\rho$, and let $\rho_c(\lambda)$ be its critical density f…

Let $\mathcal{C}_p$ be the class of probability distributions on $\Omega$ that are independent and identically distributed over the vertices, isotropic, and have expected degree eq…

Let $L$ be an irreducible dispersal matrix, let $A_i=R_i+L$ be the matrices in the differential inclusion model, and let $x_i$ be eigenvectors of $A_i$. Assume that the eigenvector…

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…

Simplified-model local limit conjecture. For all sufficiently large $n$, for the upper extremal point $M_n^+$ and every integer $k$,

Asymptotic shape conjecture. There exists a bounded convex set $U\subset\mathds{R}^d$ such that, for every $\varepsilon>0$,

Let $\tau_{0}$ be the relaxation time at the origin in the mixed threshold FA model, let $q$ be the vacancy density, let $\nu$ denote the law of the random environment, and let…

Let $p$ and $q$ be symmetric unimodal weight functions, meaning that $i\mapsto p(i)$ and $i\mapsto q(i)$ are symmetric and decreasing for $i\geq0$. Let $p\preceq_Mq$ denote majoriz…

A nonadaptive random walk in changing environment (nonadaptive RWCE) is a random walk whose edge weights are chosen in advance and therefore do not depend on the walker's location.…

Consider a labyrinth evolving by adding edges between adjacent vertices of $\mathbb{Z}^2$, without removing existing edges, and let Theseus perform the random walk on the resulting…

Let $\alpha\in(0,1)$ be the exponent governing the randomly biased random walk. Aging conjecture. In the case of randomly biased random walks, aging similar to the result establish…

Let $\alpha$ be the exponent defined in the paper's theorem on randomly biased random walks. Quenched central limit theorem conjecture. For randomly biased random walks, a quenched…

Consider random walks in environments on ${\mathbb{Z}}$ composed of two types, $(p,1-p)$ and $(q,1-q)$, where $\frac{1}{2}<p<q<1$. Equally spaced drifts conjecture. An environment…

Let the background process have a unique stationary distribution. For each environment state $i\in\{1,2\}$, let $c_i(x,\eta)$ be the flip rate at site $x\in\mathbb{Z}$ for configur…