160 problems

Consider the Tree Builder Random Walk in which, at time step $n$, a new leaf is attached with probability $p_n=n^{-\gamma}$, where $\gamma$ is a real parameter. For…

Width law of large numbers. There exists a constant $W>0$ such that, as $n\to\infty$,

Persistent hub conjecture. If

Let $T$ be the random tree under consideration, let $t$ denote its size parameter, and let $\operatorname{cov}_\ell(T)$ be the minimum number of paths of length at most $\ell$ need…

Let $\mathcal D_n$ be the disordered random full binary tree with $n$ leaves, and let $N_t(\mathcal D_n)$ denote the number of occurrences of a fixed full binary tree $t$ as a frin…

Let $T_n$ be a uniformly random binary tree of size $n$, and consider the unordered trees represented by the fringe subtrees of $T_n$. The number of distinct such unordered trees i…

Expected-time asymptotics conjecture. The expectation satisfies

Let $T_d$ be the homogeneous tree in which every vertex has $d+1$ neighbors, and consider the spatial contact process with mutations on $T_d$, where $\rho$ denotes the mutation pro…

Let $T_n$ be a conditioned Galton--Watson tree with $n$ nodes, whose edges carry independent identically distributed integer-valued increments $\eta_e$ with mean $0$, finite varian…

Variance-positivity conjecture. For every other finite tree pattern $\mathcal{M}$, one has

Let $n$ be the number of vertices in a uniform attachment tree, and let the noisy edge probability be $o(\log n/n)$. A confidence set for the root vertex is optimal if it has the b…

Jordan lower-tail transition conjecture. The lower tail of the Jordan-$2$ ordering undergoes a transition from polynomial decay to stretched-exponential decay, and a similar phenom…

Early-vertex statistic conjecture. Replacing the tree-specific notion of centrality with another statistic that identifies early vertices, and studying analogous subsequent local s…

Central-vertex proportion conjecture. As $K\to\infty$, this proportion converges to

Let $\mathcal T_{\boldsymbol{n}_\kappa}$ be a random tree with prescribed degree-count vector $\boldsymbol{n}_\kappa$, and let $N_{m_\kappa}(\mathcal T_{\boldsymbol{n}_\kappa})$ de…

Let $\boldsymbol{p}$ be the limit distribution satisfying Condition1, let $\boldsymbol{n}_\kappa$ be the prescribed degree-count vector, and let $T_\kappa$ be a tree with…

Let $\mathcal{T}_n$ be a critical Galton--Watson tree with offspring distribution $\xi$ satisfying the standard extinction, criticality, and span-one conditions, and let…

Let $T_n(f)$ be the preferential attachment tree on $[n]$ generated by the nondecreasing attachment function $f:{\mathbb{N}}_0\to\mathbb{R}_{>0}$. Write $D_n(f)$ for the graph dist…

Let $\mu$ have finite $(2+\kappa)$th moment, let $\mu'$ have infinite second moment, and let $\tau_n$ and $\tau'_n$ be the corresponding conditioned random trees. Write…

Let $\mu$ and $\mu'$ be offspring distributions satisfying … with $1<\alpha\leq\alpha'<2$ and $c,c'>0$, and let $\tau_n$ and $\tau'_n$ be the corresponding conditioned random trees…

Let two independent random recursive trees have size $n$, and let their largest common subtree be the largest subtree occurring in both trees. Near-linear common subtree conjecture…

Let $mu=mun satisfy … and let$\boldsymbol{T}_n be the height-biased random tree of size $n$. Write $deg{boldsymbol{T}n}(varnothing)$ for the number of children of its root. **Root…

An ICRT is an inhomogeneous continuum random tree with parameter $ctheta$. Mixtures of ICRTs are obtained by first sampling $ctheta$ according to a distribution and then, condition…

Consider the $cmathbb R$-tree $(cmathcal{T},d)$ obtained from the line-breaking construction with intensity process $ctilde{\sigma}$, together with its limiting uniform measure…

Let $f_{\mathtt{ISE}}$ be the random differentiable density of the Integrated SuperBrownian Excursion arising as the scaling limit of the vertical profile of suitable conditioned r…