13 problems

Let $\mathcal{X}$ be a Banach space. Let $A\colon\Omega\to\mathcal{L}(\mathcal{X})$ be a generator of a semigroup $e^{At}$ such that $\lVert A\rVert$ is bounded, and let…

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…

Fix $d\geqslant 1$ and a parameter $\theta$ such that $2\theta>\frac{d-1}{2}$. Consider the inverse-Wishart polymer with delta boundary condition, defined by the step boundary cond…

In bond percolation on growing domains of the lattice, consider the giant component in the supercritical regime and its rescaled isoperimetric profile, namely the minimal boundary…

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 $f$ be the density and let $h$ and $\alpha$ be the functions defining the peeling layers. For $t$ fixed, set … Let $N_{n(\lambda,t),0,\lambda}$ denote the number of Poisson poi…

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…

Let finite-time average velocities be those associated with the random discrete dynamical systems under consideration, and let the local velocities be random with the setting descr…

Let $N\geq 1$ be fixed, and draw a sample of size $N$ from the population described in the paper's probability space. Let the sample quantities be estimators of the corresponding p…