9 problems

Let $G$ be a planar graph, and consider i.i.d. Bernoulli site percolation at parameter $p=\tfrac{1}{2}$. Benjamini–Schramm Conjecture 8. If percolation occurs almost surely, meanin…

Wieland–Wilson's conjecture. The variance of the winding angle satisfies

For a planar FK-percolation model with cluster-weight parameter $q\geq 1$, let $p_c$ denote its critical parameter and let … denote the self-dual point. Self-dual critical-point co…

Let $f$ be the random plane wave, the smooth stationary centred Gaussian field on $\mathbb{R}^2$ with covariance kernel … where $J_0$ is the zeroth Bessel function. Random-plane-wa…

Let $f$ be a planar stationary Gaussian field with covariance function $\kappa$, and suppose quasi-independence means that well-separated percolation events on domains of scale $R$…

Let $Q$ be a quad and let $\alpha$ be a smooth curve. For a discrete percolation measure $\mu_\eta$, let $N_\eta=N_\eta(Q,\alpha)$ be the minimal possible number of intersections w…

Let $Q$ be a quad and let $\alpha$ be a smooth curve. A crossing of $Q$ is a percolation crossing between the designated boundary arcs of the quad. Finite intersection property. In…

A scaling limit is any subsequential continuum limit of critical planar percolation configurations as the mesh tends to zero. Tsirelson's conjecture. Any scaling limit of critical…

Super-strong separation lemma. Conditioned on at least four of these interfaces reaching $B(R_1)$, the set of interfaces reaching $R_1$ has positive quality with positive probabili…