5 problems

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…

Let the $O(n)$ loop model be defined on the semi-infinite honeycomb lattice with the boundary oriented as in Figure(b). For $n\in[-2,2]$, assign fugacity … to occupied vertices and…