149 problems

Subcritical concentration conjecture. The size of the largest component is concentrated within a factor $1+o_{\rm p}(1)$ also in the subcritical case.

Limiting proportion conjecture. Given that the first and second epidemics are both large, $n^{-1}Z_n^{(2)}$ converges in probability to $z$ as $n\to\infty$.

Proportional-growth conjecture. The number of contaminated cells is asymptotically proportional to its expectation:

Deterministic-asymptotic conjecture. $\#\mathbb{G}_n^*$ has a deterministic asymptotic, with the asymptotic form depending on these three subdomains.

Integrability-condition conjecture. The condition $\mathbf{E}Z_{1}\log Z_{1}<\infty$ can be dropped in all of the paper's theorems.

Consider a particle system in $\mathbb{R}^{d}$ whose particles move according to a symmetric $\alpha$-stable Lévy process and undergo critical finite-variance branching at rate…

Let $(X_n)_{n\geqslant0}$ be the iterated Galton–Watson process with offspring distribution $(p_k)$ and thinning parameter $\vartheta$, where $p_0=0$, $p_1\neq1$, and $\vartheta=1$…

Let $B_k(x)$ be the functions defined for $k\geqslant1$ and $x\in(0,1]$ in the branching-process setting above. Monotonicity conjecture. Every function $x\mapsto B_k(x)$ is decreas…

Consider the clan of ancestors associated with a space-time point $(x,0)$ in the one-dimensional loss network, and let $N({\bf A}^{(x,0)})$ be the total number of rectangles in tha…

Let $q\ge 1$. Write ${\cal R}^1_{p,q}$ for the set of probability measures under consideration, and let $p_{\text{c}}^0(q)$ and $p_{\text{c}}^1(q)$ denote the corresponding critica…

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…

Boundedness conjecture. The condition

Modified-process equivalence conjecture. For some class of functions $f$, the asymptotic estimates obtained from the modified and unmodified weight systems or branching processes a…

Type III branching-process conjecture. For type III triangulations, one can construct $\zeta_3$ and $\bar{\zeta}_3$ so that the analogue of the branching-process theorem holds. The…

Let $(S_\ell)$ be the special sequence described at the beginning of the section, and let $\mathfrak F_\ell$ be the random labelled forests associated with the coalescing partition…

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…

R-star configuration conjecture. This combination procedure should continue in further branching events: configurations resulting from collections of ancestral interaction graph re…

Let $\alpha$ denote the parameter governing the wave regime and let $N$ be the population-size parameter. Relevant time-scale conjecture. For $\alpha \in [3/2,1)$, the relevant tim…

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…

Let $L$ be the elliptic operator considered by Berestycki and Rossi, and let $lambda1'(L)$ and $lambda1”(L)$ denote their two generalised principal eigenvalues. Berestycki–Rossi'…

Independence conjecture. The random intervals $\mathcal{I}_s(u_1,u_2)$ and $\mathcal{I}_z(v_1,v_2)$ are independent; equivalently, the copula density satisfies

Let $s$ and $\lambda$ be the parameters of the generalized Poisson offspring distribution, let $P_{\rm{GP}}^\infty$ be its extinction probability, and let $P_{\rm{GP}}^{(n)}$ and…

Global fractional-linear bound conjecture. The inequality

Let $p$ denote the dispersion parameter of a random bipartite geometric (RBG) graph, and let the branching-process lower bound be … Here $f$ is a general monotone decreasing connec…

The MMM denotes the model with multiple mutations, and the FMM denotes the model with fixed mutations. A type I tail function is assumed, with the claim restricted at least to…