50 problems

For each $n\ge 3$, let $\mathcal{C}_n$ be the set of unrooted binary trees with leaves labelled by $\{1,\ldots,n\}$. Consider the Markov chain that, at each step, removes a uniform…

For the discrete Derrida–Retaux models in the critical regime, after the model-specific rescaling of time and state, the resulting processes converge in distribution, in the releva…

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…

Scaling-limit conjecture. There exists a random compact metric space $\mathcal{L}^{(S)}_{\delta}$ such that

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 $\tau^{*}$ be a sub-critical and non-generic marked Galton–Watson tree with offspring distribution $\mathbf{p}$ satisfying the condition in … . Let $M(\tau^*)$ denote the total…

Joint-growth height conjecture. As $\min(n,m)\to\infty$,

Let the PAVD model be the preferential attachment model with vertex death from Definition. Suppose that the attachment and death-rate sequences $b$ and $d$ satisfy Assumptions and.…

Immunisation conjecture. If $\alpha\geq 1$, then for every sufficiently small $\kappa>0$ there exists $\nu'=\nu'(\kappa,\eta)$ such that

Fast-speed comparison conjecture. Under these assumptions, $\lambda_1>0$ if and only if $\lambda_1^p>0$, and $\lambda_2>0$ if and only if $\lambda_2^p>0$.

Universal height lower-bound conjecture. For any permuton $cmu$ and $cvarepsilon>0$,

Maximum-leaf-height constant. The constant in the asymptotic relation $D_n^* \sim c\log n$ in probability is

Zero-boundary extension conjecture. The two-sided local-limit conjecture also holds with the cutoff parameter set to $\epsilon=0$.

Two-sided local-limit conjecture. The conditioned discrete process converges in distribution in $D([0,t_1],\mathbb{R}^3)$ to $\xi^\star$, started at $(\delta_0,m_0,s_0)$ and condit…

Fix parameters $(\lambda,s)$ in the correlated Erdős–Rényi model $\mathcal{G}(n,\lambda/n,s)$. One-sided correlation detection in trees is said to fail when none of the equivalent…

Let ${\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. For $n \ge 0$ with $Z_n({\mathrm{w}})>0$, let ${\mathcal{T}}_n$ be a simply…

Let ${\sf TDLA}_0(n,[0,+\infty)^2)$ be the square DLA tree with $n$ vertices, rooted at the corner ${\bf c}$, and let ${\bf v}$ be a uniformly sampled vertex of the tree. Write…

Let ${\sf TDLA}_0(n,{ \sf Torus}(n^c))$ denote the finite-graph DLA tree with $n$ vertices rooted at $0$, and let ${\bf dla}_n$ denote the standard DLA tree with $n$ vertices. The…

Limiting energy conjecture. For every $\alpha$, there exists $g(\alpha)$ such that, almost surely,

Consider TASEP with a reservoir of rate $\lambda=\rho q$ for some $\rho \in (0,\infty)$, and suppose that a flow rule holds for a flow of strength $q$. Let $\pi_{\lambda}$ be the e…

Let $0 < \gamma \leq \alpha < 1$. Consider the unlabelled $(\alpha,\gamma)$-chain on rooted multifurcating trees with $n$ leaves, obtained by selecting and deleting a uniform leaf…

Let $\mathcal{T}$ be a $(\mu,f,\ell)$-RIF tree, and define … Assume that $\alpha=\infty$, and let $N_0(t,B)$ denote the number of degree-zero vertices present by time $t$ whose fit…

Let $\mathcal{T}$ be a $(\mu,f,\ell)$-RIF tree, and define … For $\alpha<\infty$, let $N_k(t,B)$ denote the number of degree-$k$ vertices present by time $t$ whose fitness lies in…

Expected-time asymptotics conjecture. The expectation satisfies

Total-length fluctuation conjecture. As $n\to\infty$,