Critical level-set percolation on finite graphs and spectral gap

We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter $h \in \mathbb{R}$. We characterize the near- and off-critical phases of this model for any expanders family $\mathcal{G}_n = (V_n, E_n)$ with uniformly bounded degrees. In particular, we show that the volume of the largest open cluster at level $h_n$ is of the order $|V_n|^{\frac23}$ when $h_n$ lies in the corresponding critical window which we identify as $|h_n| = O(|V_n|^{-\frac13})$. Outside this window, the volume starts to deviate from $\Theta(|V_n|^{\frac23})$ culminating into a linear order in the supercritical phase $h_n = h < 0$ (the giant component) and a logarithmic order in the subcritical phase $h_n = h > 0$. We deduce these from effective estimates on tail probabilities for the maximum volume of an open cluster at any level $h$ for a generic base graph $\mathcal{G}$. The estimates depend on $\mathcal{G}$ only through its size and upper and lower bounds on its degrees and spectral gap respectively. To the best of our knowledge, this is the first instance where a mean-field critical behavior is derived under such general setup for finite graphs. The generality of these estimates preclude any local approximation of $\mathcal{G}$ by regular infinite trees -- a standard approach in the area. Instead, our methods rely on exploiting the connection between spectral gap of the graph $\mathcal{G}$ and its connection to the level-sets of zero-average Gaussian free field mediated via a set function we call the zero-average capacity.