25 problems

Let $\kappa$ satisfy condition … be the fixed-point equation for the corresponding branching-process generating function. Necessary-and-sufficient condition for subcriticality. One…

Bouncing-ray conjecture. If $\mu$ has mean zero and finite variance and $\Gamma$ almost surely has bouncing rays, then $\Gamma$ almost surely has escaping rays. This is presented a…

Let $G_n$ be the random graph and let $o_1,o_2$ be the two vertices whose distance is considered. Assume the hypotheses of Theorem. Then typical distance lower-bound conjecture. Th…

Threshold conjecture. For all such $\gamma$ and $\nu$: if $\mathrm{R}_{0}(\gamma,\nu)<1$, then the $(\gamma,\nu)$-EpiSLFV process goes extinct; if $\mathrm{R}_{0}(\gamma,\nu)>1$, t…

Let $0<\alpha\leq 1$ and consider the allometric setting, with $m_{x_0,R}(\xi_0)$ denoting the mean number of offspring from initial state $x_0$, radius parameter $R$, and initial…

Let $\tilde h^\tinfty$ denote the invariant spatial distribution of particles, and suppose that, for some slowly varying function $\ell$ and $\alpha\backslashin(1,2)$, condition … h…

Fix $s>1$. In the modified iteration procedure, each hole of perimeter $m$ is filled using a ring whose inner boundary has length either $\lfloor sm\rfloor$ or…

Let $T(\clock)$ be the branching process tree, and let … be the $k$-neighbourhood of the invasion percolation cluster. Under the conditions of the volume-growth theorem, let…

Let $\rho_2$ be the threshold separating the weakly pushed and fully pushed regimes, and let $(\overline N_t)_{t\geq 0}$ denote the rescaled particle-number process. Fully pushed F…

Let $\rho_1=1+\frac{\pi^2}{4}$, and let $(\overline N_t)_{t\geq 0}$ denote the rescaled particle-number process. Pulled-regime CSBP conjecture. If $\rho<\rho_1$, under suitable ass…

Let $G=(V,E)$ be a finite directed graph. For each vertex $v\in V$, let $S_v$ be a finite state space, let $P_v$ be an irreducible stochastic transition matrix, and let $R_v$ be a…

Let $S$ be the environment state space, let $f(M)=\sum P_nM^n$ be the matrix generating function, and let $E$ be the extinction matrix. Write $I$ for the identity matrix, and let…

Let $\{(Z_n,\widetilde{Z}_n)\}_{n\in\mathbb{N}_0}$ be a PP-2SDDBPO, with total population variables $T_n$ and $\widetilde{T}_n$, parameter $\mu$, and positive initial values…

Limiting composition conjecture. Suppose

Critical phase-transition conjecture. The expected number of cars that can leave the root, $X$, satisfies

Long-term behavior dichotomy. The following dichotomy holds:

Let $\mathbb{T}_{d}$ be the homogeneous $d$-ary tree with $d\geq 3$, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. The regular-tree uniform-shift transience conjecture.…

Let $\mathbb{T}_{2}$ be the binary tree, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. Define $\mathcal{A}_{\mathrm{rec}}^{\mathrm{conj}}$ to consist of all shifts of se…

Let $X$ be the canonical process under the excursion measure $\mathbf{N}$ of the strong Markov process $X-I$ at zero. Write $H$ for the height process, $\zeta$ for its lifetime, an…

Let $X$ be the canonical process under the excursion measure $\mathbf{N}$ of the strong Markov process $X-I$ at zero. Write $H$ for the height process, $\zeta$ for its lifetime, an…

Consider $N$-BBM and $N$-BRW, with empirical measures viewed under their invariant measure $\lambda^N$. Quasi-stationary distribution conjecture. Both $N$-BBM and $N$-BRW are ergod…

Geometric limit conjecture. As $t\to\infty$, on the survival event, $L_t(x_t)$ converges in distribution to a non-degenerate geometric random variable. This conjecture concerns the…

Let $d\geq 3$ and let $\varrho>0$. In the two-type symbiotic branching model $\textrm{SBM}_\gamma(\varrho)$ on $\mathbb{Z}^d$, coexistence means that there are summable initial con…

Let $X_t$ be the total quantity of parasites at time $t$, let $V_t$ be the set of cells at time $t$, let $N_t=\#V_t$, and let $X_t^i$ denote the parasite quantity in cell…

Let $Z_n$ be the population measure at generation $n$, let $g\in C_c^+\left(\mathbb R^d\right)$, and write $\langle Z_n,g\rangle$ for its integral against $g$. Let $\gamma$ be the…