60 problems

Let $v_N$ denote the asymptotic velocity of the $N$-BBM. Brunet–Derrida velocity conjecture. As $N\rightarrow\infty$, … This conjecture predicts the leading finite-population corre…

Let $Y$ be the total number of particles that have lived before extinction in a branching random walk with absorption at a barrier. The critical case is the analogue of…

Randomized-Gumbel conjecture. The maximum particle, centered by $m(t)$, converges in distribution to a randomized Gumbel.

Let $\psi$ be a selection function such that $\psi(x)\geq\delta$ for all $x\in[0,1]$ and $\psi(x)=\delta$ for all $x\in[1-p,1]$ for some $p\in(0,1)$. Asymptotic-velocity conjecture…

Let $\psi:[0,1]\to[0,\infty)$ satisfy $\psi(x)>\psi$ for $x\in[0,\epsilon]$ for some $\epsilon>0$ and $\psi(1-h)=o(h)$ as $h\to0$. Asymptotic-velocity conjecture. The $(\psi,1,N)$-…

Consider a particle system of branching Brownian motion with rank-based branching function $b:[0,1]\to[0,\infty)$ and increasing deletion function $D:[0,1]\to[0,1]$. Let $u$ denote…

Let the multitype branching Brownian motion be the model under consideration, and let condition denote the integrability condition defined earlier in the paper for the derivative m…

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

Let $N$ be the number of particles, let $\beta$ denote the branching parameter, and let $\mu_c^N$ be the critical drift for the $N$-particle Brownian bees system. For a drift $\mu$…

Glenz–Kistler–Schmidt conjecture. A law of large numbers, as stated in equation $$ , should hold for all models within the BBM-universality class.

Consider a branching Brownian motion with particle positions $X_u(t)$, alive particles $N_t$, maximum … and centering … Let $Z_\infty$ be the almost-sure limit of the derivative ma…

Derrida–Mottishaw conjecture. For any $0<a<b<1$,

Mueller–Munier conjecture. The typical size of $\mathcal{C}([-x,0])$ differs from $\mathrm{e}^{\sqrt{2}x}$ by a stretched-exponentially small factor:

Time-inhomogeneous branching random walk asymptotic conjecture. If $L(T)\sim\alpha T^{1/3}$ for $\alpha\in\mathbb{R}$, then

Critical-profile conjecture. The rescaled function $u(t,e^{c_*t}\cdot)$ converges, as $t\to+\infty$, to a nontrivial limiting function.

Consider a branching Brownian motion in the semipushed regime, with … A time-changed $\operatorname{Beta}(2-\alpha,\alpha)$-coalescent is the conjectured limiting genealogy after t…

Consider a branching Brownian motion $(\mathbf{X}_t)$ in the critical regime. Let … The BBM is called pulled when $\lambda_1^\infty=0$, semipushed when…

Mean-field and unconditioned-density conjecture. In these asymptotic limits, the fluctuations of the densities $N^{(i)}$ can be neglected, the sum over branchings can be replaced b…

Roberts–Schweinsberg conjecture. The number of particles near $z_n$ in the long run should be proportional to

Let $R_t^*$ denote the maximum modulus of branching Brownian motion at time $t$, let $m_t$ be its centering, let $\gamma^*$ be the associated constant, let $Z_\infty$ be the limiti…

Let $Z_t(F)$ be the functional of the derivative Gibbs measure associated with branching Brownian motion, and suppose that $F(x)\sim x^{-\alpha}$ as $x\to0$ for some $\alpha$ satis…

Let $\mathcal{N}_t^2$ be the set of type-$2$ particles at time $t$, with positions $X_u(t)$, and let $\overline Z_\infty$ denote the previously defined limit of the relevant multit…

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…

Selection principle for the one-dimensional $N$-BBM. The unique invariant distribution of the system seen from the leftmost particle should converge, as $N\to\infty$, to the centre…

Let $t$ be the observation time and condition the absorbed branching Brownian motion on survival up to time $t$. Let $\varepsilon$ measure the distance from criticality, as in the…