12 problems

Consider planar last-passage percolation with geometric weights. Let $T_n(\omega)$ and $T_n(\tilde\omega^t)$ denote the last-passage times in the original and site-resampled enviro…

Let $(G_n)_{n\in\mathbb N}$ be a sequence of connected graphs, with $n$ vertices, edge set $E_n$, and maximum degree $\Delta_n$. Let $t_n$ be a non-negative sequence of times, let…

Let $T_0$ and $T_t$ be the passage times at perturbation times $0$ and $t$, and let $\pi_0$ and $\pi_t$ be the corresponding geodesics. Write $\operatorname{Var}(T)$ for the varian…

Let $(p_n)_{n\geq 1}$ be a sequence of biases and let $(f_n)_{n\geq 1}$ be Boolean functions, with $C_n$ the number of changes of $f_n(X_t^{(n)})$ during $(0,1)$. A sequence…

Exceptional-time conjecture. Almost surely, exceptional times at which $\mathcal{C}_t$ contains an unbounded component exist, and the Hausdorff dimension of the random set of such…

Let $(G,\mu)$ be a random walk, where $G$ is a group and $\mu$ is its probability measure. The walk is entropy noise sensitive when its entropy noise sensitivity is defined in the…

Let $G$ be the first Grigorchuk group. The first Grigorchuk group's conjecture. The first Grigorchuk group is $\ell^1$-noise sensitive. This is presented as a particular case of th…

Noise sensitivity conjecture. For any $d greater than or equal to k 2$, the event $\mathcal{C}n$ is noise sensitive.

Voronoi noise-sensitivity conjecture. Voronoi percolation is noise sensitive at criticality.

Shape-noise-sensitivity conjecture. If $S$ is bounded and simply connected, then the Poisson Boolean model for $S$ is noise sensitive at criticality.

Random-radius noise-sensitivity conjecture. For every $R>0$ and $\mu_R$, the Poisson Boolean model with random radii chosen according to $\mu_R$ is noise sensitive at criticality.

Nearest-neighbor noise-sensitivity conjecture. If $t_n \gg (n^2\alpha_4(n))^{-1} \approx n^{-3/4}$, then