7 problems

Ratio-monotonicity conjecture. As the number of cells in $M_n$ increases, the ratio $R_n$ also increases. The paper gives this as a natural conjecture in an informal discussion, mo…

Let $M_n$ be the finite honeycomb-lattice region. For $k=1,2,3$, choose sets $A_k,B_k\subset M_n$, and let $U_k$ be the event that fluid $k$ percolates between $A_k$ and $B_k$. Gen…

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the…

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the…

Let $M_n$ be the finite honeycomb-lattice region with four-colored cells, and let $B_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates between opposite sid…

For rational magnetic flux $p/q$, let $r$ denote the gap label and let $\sigma_{\rm H}$ be the Chern number associated with the gap, satisfying the Diophantine equation … for some…

Flory–Nienhuis displacement conjecture. The mean-square displacement satisfies