11 problems

Consider the belief-propagation operator for the symmetric Potts broadcasting model, with reconstruction threshold and Kesten–Stigum threshold given by $d\lambda^2=1$. Let BP uniqu…

Consider robust reconstruction on a tree, where the leaves are observed through identical noisy observation channels, and let $h$ be the tree depth. Maximum-likelihood channel-inde…

In the broadcast-on-trees setting, let BI denote boundary irrelevance, namely the property that leaf observations become independent of the root variable when conditioned on the su…

In the broadcast model on trees, let the Kesten–Stigum threshold be the regime determined by the branching factor and the second eigenvalue of the broadcast chain, and let inferrin…

Consider the model on a regular tree of degree parameter $d$, with broadcast parameter $\theta$, and a BMS survey channel $W$. Write $P_e(W)$ for the survey error probability, and…

Let $T$ be a regular tree or a Galton-Watson tree with Poisson offspring distribution, with root vertex $\rho$. Let $(T,\rho,\theta,W)$ denote the corresponding broadcast model wit…

Let $P_e(\delta)$ denote the optimal reconstruction error probability in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write…

Let $I(delta)$ denote the mutual information in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write $\delta=\delta_c-\tau$ with…

Tree-optimality conjecture. $P_G(a,b)=P_T(a,b)$.

Let $T$ be an infinite tree with root $\rho$. Assign the root a uniformly random label in $\{\pm1\}$ and propagate labels along edges, retaining the parent label with probability…

Let $T$ be an infinite tree with root $\rho$. In the two-state broadcast process, each vertex label $\tau_v\in\{\pm 1\}$ is generated by assigning the root a uniformly random label…