Abstract: In two-dimensional Bernoulli percolation, we declare each edge of the square grid \(Z^2\) to be open with probability p or closed with probability \(1-p\), independently from edge to edge. There is a critical value \(p_c = 1/2\), such that for \(p < p_c\), all components of open edges are finite, and for \(p > p_c\), there is a unique infinite component of open edges. In 1983, Grimmett introduced the following variant. Let \(f\) be a nonnegative real function on \([0,\infty)\), and consider the subgraph \(G_f\) of \(Z^2\) induced by the edges between the positive first coordinate axis and the graph of \(f\). Grimmett found that if \(f(u) \sim a \log u\) as \(u \to \infty\), the critical value \(p_c(f)\) for percolation on \(G_f\) equals a specific function of \(a\). In 1986, Chayes-Chayes considered the function \(f(u) = a \log(1+u) + b \log(1+\log(1+u))\) and showed that if \(b > 2a\), then the percolation \(G_f\) has an infinite open component at the critical point (i.e., a discontinuous phase transition). In joint work with Wai-Kit Lam, we prove that the phase transition is discontinuous if and only if \(b > a\), and we compute sharp asymptotics for all \(p\), \(a\), and \(b\) of the expected passage time in \(G_f\) from the origin to the vertical line \(x = n\) in the related first-passage percolation model, improving results of Ahlberg. We also find asymptotics for the variance and a central limit theorem.