Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 5

A site $v$ is pivotal for an increasing event $E_N$ in a configuration $\omega$ when changing only the state of $v$ changes whether $E_N$ occurs. Away from the boundary, pivotality for a left-to-right crossing has the geometric four-arm description: from neighbors of $v$ there are two disjoint open arms reaching the left and right sides and two disjoint closed arms reaching the top and bottom sides, in alternating cyclic order. The boundary version truncates the corresponding arms at the sides already adjacent to $v$.

The Margulis–Russo formula gives

Set $N=3^j$. At $p=1/2$, first use the Russo-Seymour-Welsh theorem to produce, with probability bounded below independently of $N$, two separated open crossings from left to right and a closed crossing between them. Explore the interface separating the lower open cluster from the adjacent closed cluster until it reaches the opposite macroscopic boundary. The explored interface supplies two alternating arms, while fresh RSW crossings in the unexplored regions supply the other two.

Repeat this construction in the geometrically separated scale bandsThe domain Markov property of a percolation exploration leaves unrevealed sites with their original independent critical law. RSW and the Harris-FKG inequality therefore give a uniform conditional probability $c>0$ that the required open and closed connections occur in each band. On that event the exploration identifies a site having four alternating arms to the four sides of $W_N$, hence a pivotal site for $E_N$.

Choose the candidates in disjoint scale bands, so successful bands give distinct pivotal sites. If $Y_k$ indicates success at scale $k$, thenThusfor $j\ge 2$, after adjusting the positive constant. This is the required logarithmic-in-$N$ lower bound.