10 problems

Extremal-process moderate-deviation conjecture. As $n\to\infty$,

Maillard–Mallein conjecture. Any point measure $\zeta$ satisfying this invariance relation has the law of a randomly shifted decorated Poisson point process.

Mean-regime conjecture. There exists a constant $C_{\mathrm{m}}>0$ such that, for every $y\in\mathbb{R}$,

Exponential-population branching random walk speed conjecture. As $T\to+\infty$,

Let $\beta>1$ and $\sigma^2=\frac{\beta}{2\beta-1}$, and let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$. Write $Z^{(1)}_\infty$ for the…

Let $d\geq 1$, let $\mathcal{N}_t$ be the particles at time $t$, and let $X_t(u)\in\mathbb{R}^d$ be the position of particle $u$. Define … and, for $u\in\mathcal{N}_t$, define its…

Let $Z(dx)$ be the random measure appearing in the limiting extremal process of the two-dimensional discrete Gaussian free field (DGFF), whose intensity is $e^{-2x}\otimes Z(dx)$.…

McKean's conjecture. When $\sigma_1=1$, this martingale should be the martingale appearing in the representation of the extremal distribution of branching Brownian motion.

Extremal-process conjecture. In the limit of large times, the distributions of $Xi(t)$ and $Pir$ coincide:

Brunet–Derrida conjecture. The extremal process of BBM is invariant under superposition: the collection of finitely many independent copies, with possibly relative shifts, has the…