71 problems

Uniqueness and convergence conjecture. Under mild conditions, for every $q\in[0,1]$ and every $m$, there is a unique nontrivial fixed point, and an initial distribution converges t…

COA maximum-principle conjecture. For certain COA models, the above characterization is valid with an appropriate function $g$.

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

Let $\mathcal{T}^{(n)}_{\ell}$ denote the full tree of the sequentially Markov coalescent at genomic distance $\ell$, let $\Pi_n$ be its stationary distribution, and let…

Macroscopic-event jump conjecture. The macroscopic events captured by $\Psi$ should give rise to jumps in the forward-in-time diffusion process.

Let $N$ be the population size and consider a suitably adjusted version of the diploid Moran model described in the source, with overlapping generations and no selfing. Let the mod…

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

Applicability conjecture. The above techniques could be applied in this setting of interacting populations with fluctuating population size.

Let $n>2$ be fixed, and sample $n$ genes at time-step $0$. Let $(\Pi_k^{N,n})_{k\in\mathbb{Z}_+}$ be the ancestral process taking values in the space $\mathcal{E}_n$ of partitions…

Wakeley's polynomial stationary-distribution property. There exist constants $r_k$, for $k=1,\ldots,N$, such that

Consider the moderate-selection regime $s_N\asymp N^{-b}$ for some $0<b<1$, where $N$ is the population size and $s_N$ is the selective advantage. A contending mutant subpopulation…

Let $N$ denote the population size, and consider the Moran model specified in Section. Write $\overline v_N$ for its asymptotic speed of adaptation, analogous to the speed found by…

The model is a branching particle system for a population accumulating deleterious mutations, with typical type-distribution width $c\log(N)$ for some $c\in(0,1)$. Branching partic…

Limiting conditional genealogy conjecture. If $y_*=0$, then the conditional genealogy behaves like scenario (ii); if $y_*\in(0,1]$, then the conditional genealogy behaves like scen…

Let $\mathcal I$ be a randomly sampled individual, and for $k\in\mathbb N_0$ let $\mathscr E_k$ be the event that there is a coalescence event between $\mathcal A_k$ and…

Consider a stochastic population model with logistic growth and nonlinear diffusion corresponding to the porous medium equation, in which cooperative behaviour is captured by the n…

Let a population expand into new territory as a travelling wave, and consider genealogies in a model with local population density tending to infinity. Bolthausen–Sznitman genealog…

Conjecture. Theorem 1 remains valid under the additional assumption that $\rho_k<\infty$ for all $k\in E$.

Optimality conjecture. There are constants $c_x>0$ and $\xi>0$ such that

Let $k$ ancestral lineages be sampled from near the wavefront, and let $\pi$ be the stationary distribution of a single ancestral lineage relative to the wavefront. Assume that the…

Let $B$ be the wave parameter and let $b1$ denote the corresponding parameter, with $B=2$ corresponding to $\alpha=3/2$. Consider the genealogical trees of a sample of $A$ alleles…

Consider the host model and its results reviewed from the cited work, in which hosts have i.i.d. standard exponential lifetimes. Let the hosts' lifetimes instead be i.i.d. random v…

Let $N$ be the population size, and let $b$ be the exponent governing the scaling of the selection parameter, with … Let Theorem denote the law of large numbers for the relative fi…

Let $N$ be the population size, and suppose the mutation and selection parameters scale as … where $0<b<\frac{1}{2}$ and $a>3b$. Let Theorem denote the law of large numbers for the…