5 problems

Tóth's conjecture. The quantity

Let $G$ be a finite weighted hypergraph with non-negative weights. At each step, choose a hyper-edge with probability proportional to its weight and then choose a uniformly random…

Tóth's conjecture. Almost surely, $\pi_\beta$ has only finite cycles when $d=2$, for every $\beta>0$, whereas when $d\geq3$ it has infinite cycles for every $\beta>\beta_c$.

Let $k\geq 2$, let $\mathcal S_k$ be the symmetric group on $k$ letters, and let $g:\mathcal S_k\to\mathbb R$ be any function. For nonnegative numbers…

Let $X_N=\{0,1\}^N$ be the vertex set of the $N$-dimensional hypercube, let $X_{N,2}=\{\{i,j\}:1\leq i<j\leq N\}$ be the set of coordinate pairs, and let $q:X_{N,2}\to[0,\infty)$ b…