25 problems

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 $t(n,k)$ denote the rectangle crossing time, and let $(k_n)_{n\ge1}$ be a sequence with $k_n\to\infty$. Being above the median of $t(n,k_n)$ means the corresponding indicator p…

Let $T_k^n$ denote the relevant point-to-point passage time, and let $\operatorname{Var}(T_k^n)$ be its variance. Variance conjecture. The variance satisfies … This prediction is u…

Let $(f_n)_{n\geq 1}$ be Boolean functions on $\\{0,1\\}^n$, and let $(X^{(n)})_{n\geq 1}$ be continuous-time $p_n$-biased random walks, with $C_n$ denoting the number of value cha…

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…

Dynamical exclusion-sensitivity threshold conjecture. If

Let a height process arise from a polymer model with a multilinear chaos expansion in an underlying environment field, and let $F$ be a bounded measurable functional of the height…

Consider a last-passage percolation model with a perturbation parameter $t$, in which stability and chaos refer respectively to persistence and decorrelation of the geodesic, while…

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…

Jacka–Warren–Windridge conjecture. Choosing a variable with maximum current value stochastically dominates every other strategy for general $n$.

Let $H$ be the sparse random matrix and let $q$ denote its sparsity parameter. Theorem 1 and Theorem 2 concern the noise sensitivity results established in the paper for the top ei…

Benjamini–Kalai–Schramm conjecture. Knowing the Voronoi tessellation, but not the colouring of its cells, typically gives almost no information about whether the colouring contains…

Let $\mathcal G(n,1/n)$ be the Erdős–Rényi random graph at criticality, and let $\mathcal C_1$ denote its largest component. For fixed $a>0$, consider the property…

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…

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…

Benjamini–Kalai–Schramm conjecture. With high probability, the quenched crossing probabilities are very close to the annealed crossing probabilities.

Consider Voronoi percolation in a region $S$ with a point configuration whose points are independently perturbed according to Brownian motions run for time $t=t(n)$. Let the event…

Expander noise-insensitivity conjecture. The condition $d<2k+h_E$ should imply that complete occupation is noise insensitive.

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.

Spectral–pivotal singularity conjecture. The scaling limits of

Let $f_n$ be the Boolean crossing functions under consideration, with spectral sample $\mathscr{S}_{f_n}$ and pivotal set $\mathscr{P}_{f_n}$. Spectral-sample singularity conjectur…