11 problems

Pitman's Brownian continuum random tree scaling-limit conjecture for discrete tori
arXiv@arxiv
Let $K_m$ be the complete graph on $m$ vertices, let $\mathcal{T}$ be its uniform spanning tree, and let $d_{\mathcal{T}}(x,y)$ denote the distance between vertices $x$ and $y$ in…
Conjectureopencontinuum-random-trees discrete-tori uniform-spanning-trees
1
The Laplace-exponent criterion for boundedness of inhomogeneous continuum random trees
arXiv@arxiv
Boundedness conjecture. The condition
Conjectureopenbranching-processes continuum-random-trees levy-processes
1
Aldous–Pitman–Miermont conjecture on Lévy trees as mixtures of ICRTs
arXiv@arxiv
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…
Conjectureopencontinuum-random-trees levy-processes random-trees
0
The Luczak–Winkler leaf-growth conjecture for binary trees
arXiv@arxiv
Luczak–Winkler leaf-growth conjecture. This process converges to a non-trivial CRT dynamic $(\mathcal{LW}_t,t\geqslant 0)$ .
Conjectureopencontinuum-random-trees random-trees stochastic-processes
0
Continuum random tree conjecture for size-biased cluster volumes
arXiv@arxiv
Consider critical hierarchical percolation with size-biased cluster $K_n$ , viewed as a metric measure space after suitable rescaling. Continuum-random-tree conjecture. The size-bia…
Conjectureopencontinuum-random-trees hierarchical-percolation
0
The ICRT scaling-limit conjecture for trees with a given degree sequence
arXiv@arxiv
Let $(p^n)_{n\geq 1}$ be a sequence of degree-sequence parameters with associated scaling parameters, and let $\beta=(\beta_i)_{i\geq 1}$ denote the limiting hub-size parameters. W…
Conjecturesolvedcontinuum-random-trees random-trees scaling-limits
0
The unique-minimum conjecture for iterated Brownian-snake labels
arXiv@arxiv
Unique-minimum conjecture. Almost surely, ${\boldsymbol\ell}^{(j)}$ reaches its minimum only once, at a leaf of ${\bf h}^{(j)}$ .
Conjectureopenbrownian-snakes continuum-random-trees stochastic-processes
0
Aldous–Miermont–Pitman boundedness criterion for inhomogeneous continuum random trees
arXiv@arxiv
Aldous–Miermont–Pitman conjecture. For $\boldsymbol{\theta}\in\Theta$ , the tree $\mathcal{T}_{(\infty)}^{\boldsymbol{\theta}}$ is bounded if and only if
Conjectureopencontinuum-random-trees random-trees
0
The Poisson-snake conjecture for limiting chronological height processes
arXiv@arxiv
The limiting genealogical structure is encoded by a continuous path, and the chronological height process is obtained by marking the branches of the forest with a Poisson point pro…
Conjectureopenbranching-processes continuum-random-trees scaling-limits
0
Ergodicity conjecture for the Branch Merging Markov Chain
arXiv@arxiv
BMMC ergodicity conjecture. The Brownian CRT is a stationary distribution of the BMMC, and, for any continuum tree as initial state, the BMMC converges in distribution to the Brown…
Conjectureopencontinuum-random-trees gromov-hausdorff-topology markov-chains
0
Time-change conjecture for the CRT pruning process and the beta(3/2,1/2)-coalescent
arXiv@arxiv
Aldous's continuum random tree pruning yields an interval-partition-valued process, and the paintbox procedure associates such a process with a partition-valued coalescent. The pro…
Conjectureopencoalescent-processes continuum-random-trees random-trees
0