4 problems

At criticality, let $C(0)$ be the cluster containing the origin, condition on $|C(0)|=n$, and scale space by $D^{-1}n^{-1/4}$, where $D$ is the model-dependent scaling constant. Le…

Let $f$ be the density of integrated super-Brownian excursion (ISE), and let $F(t)=\int_{-\infty}^{t}f(s)\,ds$ be its distribution function. Chapuy–Marckert's conjecture. The deriv…

Bousquet-Mélou and Janson's conjecture. The derivative $f'_{ISE}$ possesses a continuous version, but no second derivative.

Second-order diffusion conjecture. We conjecture that $f_{ISE}$ can be represented as a second-order diffusion of the form