30 problems

Moment lower-bound conjecture. For any $r\geq 2$, the same inequality holds for every $\alpha\in(0,r-1]$. This conjecture concerns lower bounds for the critical probability of boot…

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 $\tau^{*}$ 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…

Distance-profile width moment conjecture.

Small-component conjecture.

Simply generated forest conjecture. Lemma should also hold when the generating function has radius of convergence $0$.

Transience conjecture. There exists a constant $c=c(\mathsf{GW})>0$ such that $\mathrm{FM}(\mathsf{T},\eta,\mathsf{SRW})$ is transient for $\mathsf{GW}$-almost all infinite realiza…

Subcritical flux conjecture.

A Galton–Watson tree is equipped with i.i.d. exponential edge lengths, and its combinatorial shape is sampled from an invariant Galton–Watson measure $\mathcal{IGW}(q)$ introduced…

Let $cmu$ be a degree distribution satisfying the giant component condition, and let $ctilde{\mu}$ be its size-biased distribution. Write $clambda_c^-(\mu)$ and $clambda_c^+(\mu)$…

Let a Galton–Watson tree be supercritical and condition it on survival. Consider the simple exclusion process on this tree in the constant speed model and in the variable speed mod…

Let $[\boldsymbol{T},\boldsymbol{o}]$ be a unimodular eternal Galton–Watson tree. Suppose its offspring distribution is in the domain of attraction of an $\alpha$-stable distributi…

Let $d_{\mathrm{rec}}^{\mathrm{graph}}$ denote the threshold for reconstruction on the random graph, and let $d_{\mathrm{rec}}^{\mathrm{GW}}$ denote the reconstruction threshold on…

Let $m$ be the offspring mean of the supercritical Galton–Watson tree, let $\mu_{\mathsf{HARM}_\lambda}$ denote the harmonic-measure environment law, and let $\mathbf{AGW}_\lambda$…

Let ${\mathbb T}$ be a supercritical Galton–Watson tree with offspring mean $m$, let $\Xi_\lambda$ be the $\lambda$-harmonic ray, and let $(X_n)_{n\geq 0}$ be the $\lambda$-biased…

For integers $m,n_1,\dots,n_m$ and parameters $\nu_{1}^{(1)},\dots,\nu_{n_1}^{(1)},\nu_{1}^{(2)},\dots,\nu_{n_2}^{(2)},\dots,\nu_{n_m}^{(m)}$, consider the functions $g_{\xi}$ aris…

Let $T$ be a supercritical Galton–Watson tree without leaves, and consider biased random walk on $T$ with bias pointing away from the root. Let $v(\lambda)$ denote its velocity as…

Differentiability conjecture. The density $\widehat f$ is not third-order differentiable at any integer $n\geq 2$.

Forest-fire cluster-shape conjecture. In this forest-fire model, typical clusters are again subcritical Galton–Watson trees before gelation and critical Galton–Watson trees past ge…

Typical-cluster conjecture. Past gelation, typical clusters are again $cP(1)$ Galton–Watson trees.

Conjecture on arbitrary subtrees. The average size of an arbitrary subtree is asymptotically normal, as shown by Wagner in the corresponding uniform random labelled-tree case.

Conjecture on increasing toll functions. Similar asymptotic results should hold for general conditioned Galton–Watson trees and increasing $f$, but the precise limits presumably de…

For $r\geq 2$, let $f_r^{GW}(b)$ denote the relevant extremal critical probability for Galton–Watson trees, and let $c_r,C_r$ be positive constants depending only on $r$. Sharpness…

Let $r e 2$ and $b e r$ be parameters, let $theta{r,b}$ denote the family of offspring distributions constructed in the proof of Lemma, and let $fr^{GW}(b)$ denote the infimum of…

Lyons's conjecture. The speed $v(\beta,P)$ increases in $\beta$ on $(1/E[Z],\infty)$.