7 problems

Mézard–Montanari's conjecture. If $q\leq 4$ and $d<d_{\max}$, then there is reconstruction if and only if

Let $q\ge 1$. Write ${\cal R}^1_{p,q}$ for the set of probability measures under consideration, and let $p_{\text{c}}^0(q)$ and $p_{\text{c}}^1(q)$ denote the corresponding critica…

Potts monotonicity conjecture. If reconstruction is solvable for $M_1$, then it is also solvable for $M_2$.

Consider the symmetric Potts channel $M$ on $q\geq 3$ symbols, with $b$-ary tree branching factor $b$ and second eigenvalue $\lambda_2(M)$. Potts-model below-bound conjecture. Ther…

Let $M$ be a channel with second eigenvalue $\lambda_2(M)$, and let $T_b$ be the $b$-ary tree. The reconstruction problem is census-solvable when the root can be reconstructed from…

Consider the Ising model on the regular tree $T_b$. A local reconstruction algorithm is one in which each vertex scans the information stored at its descendants $\ell$ generations…

Let $\mathcal{T}$ be the $d$-regular tree, with $d\ge 3$, and let $\beta$ be the inverse temperature. A probability measure is a factor of IID if it can be obtained as an equivaria…