2 added 4 characters in body

Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) direction. For each configuration $\omega \in A$ one can define pivotal edges as such open edges in $w$, closure of which seizes the crossing. Let us call a pivotal edge $e$ positive (resp. negative) if passing through $e$ in our LR crossing (since it is pivotal, there is only one way to do so) we go outwards (resp. towards) the origin. It is known that at criticality the expected number of all pivotal edges is of order $\log n$. The question is, is it true that positive pivotal edges dominate negative ones such that the expectation of the difference goes to infinity as well? Can one prove that it is at least positive for large $n$?

P. S. I would also appreciate a reference or an exposition of original Kesten's geometrical arguments to prove the $\log n$ estimate. It seems that modern books rely either on sharp threshold results (B. Bollobas, O. Riordan) or to exponential decay for subcritical values (G. Grimmett).

1

# More positive pivotal edges than negative ones at critical bond percolation on Z^2?

Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) direction. For each configuration $\omega \in A$ one can define pivotal edges as such open edges in $w$, closure of which seizes the crossing. Let us call a pivotal edge $e$ positive (resp. negative) if passing through $e$ in our LR crossing (since it is pivotal, there is only one way to do so) we go outwards (resp. towards) the origin. It is known that at criticality the expected number of all pivotal edges is of order $\log n$. The question is, is it true that positive pivotal edges dominate negative ones such that the expectation of difference goes to infinity as well? Can one prove that it is at least positive for large $n$?

P. S. I would also appreciate a reference or an exposition of original Kesten's geometrical arguments to prove the $\log n$ estimate. It seems that modern books rely either on sharp threshold results (B. Bollobas, O. Riordan) or to exponential decay for subcritical values (G. Grimmett).