7 problems

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…

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…

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…

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 $T_\xi$ be a Galton–Watson tree with offspring distribution $\xi$. Moment lower-bound conjecture. For every $r\geq 2$ and $\alpha\in(0,r-1]$, there exists a constant…