13 problems

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

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…

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…

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

Let $\Gamma$ be a nonelementary hyperbolic group with word metric $d$, let $\partial\Gamma$ have a visual metric $d_a$ for some parameter $a>1$, and let $\mu$ be an admissible, sup…

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

For a prime $p$, let $H(p)$ be the height of its Pratt tree: the tree with root $p$, whose children are the prime factors of $p-1$, constructed recursively, stopping at $p=2$. Let…

Trace strong-recurrence conjecture. Every BRW on almost every trace of a transient BRW is strongly recurrent.

Trace-transience conjecture. The trace of a transient BRW is almost surely transient for SRW.

Strong-recurrence threshold conjecture. Let $G$ be a graph with bounded degrees. Then the BRW is strongly recurrent if and only if

Hausdorff-dimension continuity conjecture. The Hausdorff dimension of the limit set is continuous for $m\neq 1/\rho(P)$ and discontinuous at $m=1/\rho(P)$.

Non-amenability conjecture. Let $\mu$ be the measure of the trace of a transient BRW on a URG. Then $\mu$ is non-amenable.

Let $B_n$ be the maximum at generation $n$ in the branching random walk considered in the paper, and let $b_n$ be its deterministic centering term. Branching-random-walk convergenc…