26 problems

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…

Consider the discrete-time model with $N$ particles on the line, whose positions at time $t$ are $Y_1(t),\ldots,Y_N(t)$ and evolve by … where $j_i$ and $j'_i$ are chosen uniformly…

Let $M$ denote the maximum position of a subcritical branching random walk in a random environment, and consider the cases $alpha>1$ with $varTheta(1)>0$ or $alpha\leqslant 1$. Und…

Bramson–Barral–Hu–Madaule conjecture. Under these assumptions, $M_n$ should be of order $n^b$. This is a conjectural prediction for the minimum of a branching random walk when the…

Let $R_n$ be the range and $Y_n$ the stationary comparison variable defined earlier in the paper, and let $u$ be the finite quantity … Let $\kappa$ denote the variance constant fo…

Consider the noisy $\lfloor N^\gamma\rfloor$-BRW and the corresponding genealogical process of its population as $N$ tends to infinity. In the related model, there is a critical va…

Let the noisy $\lfloor N^\gamma\rfloor$-BRW be a branching-selection particle system, with the selection regime determined by whether selection favors the fittest or the luckiest i…

Maillard–Mallein conjecture. Any point measure $\zeta$ satisfying this invariance relation has the law of a randomly shifted decorated Poisson point process.

Let $W_\infty^{(x)}$ and $D_\infty^{(x)}$ denote the truncated martingales started from position $x$, and let $\kappa$ be the parameter appearing in the tail behavior of…

Equality of the tail constants. The constants $C_1$ and $C_2$ are equal. This would sharpen the established two-sided asymptotic-order estimate for the tail probability of the whol…

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

Extremal-particle separation conjecture. The property that the majority of the roughly $(\log x)x^{\frac{d-1}{2}}$ particles beyond $x$ at time $t_x$ are separated in either the fi…

Let $\mathbf{T}$ be a critical binary branching Galton–Watson tree, and let $S_{\mathbf{T}}$ be a $\mathbb{Z}^4$-valued branching random walk with centred, nearest-neighbour increm…

Local-time convergence conjecture. As $R\to\infty$, this local-time process will converge to the local time of super-Brownian motion. The source gives no precise mode of convergenc…

Clustering and star-shaped genealogy conjecture. At a typical time, the majority of the population is close to the leftmost particle, and the genealogy of the population is given b…

Let $D$ be a probability distribution on $\mathbb{Z}^d$, and define … Assume that $2\leq\alpha<4$, and consider the branching random walk and its associated random walk under the a…

In branching Brownian motion, Proposition concerns the tail of the difference between the heights of nearby particles, and Theorem gives a related probability estimate. For branchi…

Let $\varrho_{n,n-k}$ denote the point measure of particles at generation $n$ whose most recent common ancestor with the minimum particle is at least $n-k$ generations from the roo…

Bérard–Gouéré speed correction conjecture. It has been conjectured that

A random scenery assigns colors independently to the sites of the lattice, and a single defect means two sceneries that differ at exactly one point. The 2-color detection conjectur…

Let $u{n,beta}$ be the Gibbs measure on particles at generation $n$, let $u{n,beta}^{otimes 2}$ denote its product measure on two sampled particles $z,z'$, and let…

Let $Z$ be the number of particles in a branching random walk with drift $c>1$ that cross the origin for the first time. Aldous's power-law tail conjecture. The random variable $Z$…

Let $Z$ be the number of particles in a branching random walk with critical drift that cross the origin for the first time. Aldous's conjecture. … This conjecture concerns the fail…

Let $S_n$ denote the event that the branching random walk survives through iteration $n$, and let $\boldsymbol{P}_p$ denote its survival probability when the cookie probability is…

Consider a binary tree ${\mathbb T}_{\tt bs}$ rooted at $e$, with independent identically distributed Bernoulli increments $Y(x)$ of mean $p\in(0,1/2)$, and define the branching ra…