Sharp small-bias asymptotics for the critical curve

Let $d\geq 2$, let $\delta$ denote the charge bias, and let $\varepsilon_\delta$ and $\kappa_d$ be the quantities defined in the paper through the small-$\delta$ expansion of the critical curve. Sharp critical-curve conjecture. As $\delta\downarrow 0$,

$$\varepsilon_\delta\sim\begin{cases}\kappa_2\delta^4\log(1/\delta),&d=2,\kappa_d\delta^4,&d\geq 3. \end{cases}$$

This conjecture asserts that the upper bound in the small-charge-bias asymptotics of $\beta_c(\delta)$ is sharp. It is linked to a conjectural large-deviation estimate for trimmed local times; the paper does not establish it.