14 problems

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…

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

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

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…

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…

Conjecture on the minimum. As $t\to\infty$, for some constant $C>0$,

Bounded front-width conjecture. For $b7<1$, almost surely,

Let $\beta=\sigma+i\tau\in\mathbb C$, let $\rho\in[-1,1]$, and let $p_t(\beta)$ be the log-partition function per unit time for the complex BBM energy model. Let $B_1,B_2,B_3$ deno…

Let $\xi$ be a point process satisfying Assumption (regularity-1), with the associated constant $c_{\xi}$. The properties (US), (SUS) and (SDP) are the conditions defined in the pa…

Let $u^R(t,x,y):=\mathbb{P}_y(M^R_t\leq x)$, where $\mathbb{P}_y$ is the law of branching reflected Brownian motion starting at $y\geq 0$. Let $q^R_{\delta}(y,t)$ be defined as in…

For one-dimensional branching Brownian motion with a smooth diffusivity profile $\sigma:[0,1]\to(0,\infty)$ whose image is contained in a compact subset of $(0,\infty)$, let…

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