Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation

We prove that the rescaled one-point fluctuations of the boundary of the percolation cluster in the Bernoulli-Exponential first passage percolation around the diagonal converge to a new family of distributions. The limit law is indexed by the rescaled level of percolation $s\ge 0$, it is Gaussian for $s=0$ and it converges to the Tracy–Widom distribution as $s\to \mathrm{\infty }$. For a fixed level $s>0$ the width of the cluster in the limit as a function of a time parameter $t$ is of order $t^{2\mathrm{/}3}$ with Tracy–Widom fluctuations as in the discrete model.