59 problems

Brunet–Derrida conjecture. The selection mechanism shifts the speed of the population by a correction term of order

Consider a branching random walk in which the offspring variables are independent and identically distributed, independent of the parent's position, and the displacements are indep…

Let $W_n^\beta$ denote the partition function of the time-inhomogeneous branching random walk at inverse temperature $\beta$, and let $\beta_c$ be its critical parameter. In the ti…

Extremal-process moderate-deviation conjecture. As $n\to\infty$,

Benjamini's conjecture. The trace of a transient BRW on $G$ has infinitely many ends.

Positive-speed conjecture. The SRW on almost every trace of a transient BRW on a URG has positive speed.

Let $d\geq 2$, and let ${\mathtt P}_{\omega}[T(0,1)>n]$ denote the quenched probability that the branching random walk has not reached the relevant target by time $n$ in environmen…

Nonelementary fixed-point conjecture. Nonelementary fixed points could correspond to random walks $S_{n}^{(\beta _{1})}$ of both types: oscillating and non-oscillating.

Let $b\in\mathbb{N}_2$ and $\rho\in(1,e)$, and suppose that, for some sufficiently large constant $L_1>0$, uniformly for $m\in[1,d/L_1]$ as $d\to\infty$, the first passage time sat…

Miermont et al.'s moment conjecture. The moment assumption for convergence of the multitype branching random walk to the Brownian snake should be reducible from order…

Let $\Gamma$ be a non-elementary hyperbolic group generated by a finite set $S$, and let $\nu$ be a probability measure on $S$ with $\operatorname{supp}(\nu)=S$. Consider the branc…

Precise lower-deviation asymptotics. There exists a constant $C>0$, depending on the law of the branching random walk and $\hat{c}_1$, such that

Let $\{Z_n\}_{n\geq 0}$ be a critical two-dimensional branching random walk started from a Poisson random measure with intensity measure equal to Lebesgue measure on $\mathbb{R}^2$…

Maximal-displacement conjecture. The maximal displacement for a supercritical branching random walk on a transitive graph should exhibit the same behaviour; namely, if $\alpha$ is…

Consider a branching random walk in random environment on $\mathbb Z^d$: particles move as continuous-time simple random walks and branch at position $x$ at rate $\xi_x$, where…

Let $\widehat\Pi_t$ be the normalized particle occupation measure of the catalytic branching random walk on $\mathbb Z^d$, and let $\lambda$ and $\lambda_0$ denote the ordinary and…

Glenz–Kistler–Schmidt conjecture. A law of large numbers, as stated in equation $$ , should hold for all models within the BBM-universality class.

In dimensions $d\leq 3$, let the infinite snake be the branching random walk conditioned to be infinite, subsequently conditioned never to reach the origin, whenever this condition…

Let $B_x$ be the shifted unit ball at Euclidean distance $x$ from the origin, let $\tau_x$ denote the first passage time of the branching random walk to $B_x$, and let…

Exponential-population branching random walk speed conjecture. As $T\to+\infty$,

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

Generic branching random walk extension conjecture. Such results should adapt to more general, namely non-Gaussian, branching random walks.

Sparse-graph crossover conjecture. 1. If

FPT conjecture for delayed branching BRW. Conditioned upon survival, the first passage time satisfies

CREM Gibbs-measure threshold conjecture. There should exist a threshold $\beta_G\geq 0$ such that the Gibbs measure can be efficiently approximated for $\beta<\beta_G$ and cannot b…