Random independent sets in triangle-free and linear Berge-$C_4$-free hypergraphs

We study probability distributions on independent sets of uniform hypergraphs with local constraints. For a positive vertex-weight function $w$ on an $(r+1)$-uniform hypergraph, define the weighted degree of vertex $v$ by \[ d_w(v):= \sum_{e\ni v} \left( \prod_{u\in e\setminus\{v\}}\dfrac{w(u)}{w(v)