Critical level-set percolation on finite graphs and spectral gap
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| Format: | Preprint |
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2026
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| author | Goswami, Subhajit Pal, Dipranjan |
| author_facet | Goswami, Subhajit Pal, Dipranjan |
| contents | 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 $Θ(|V_n|^{\frac23})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_07802 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critical level-set percolation on finite graphs and spectral gap Goswami, Subhajit Pal, Dipranjan Probability Combinatorics 60K35, 82B43, 60G60, 05C80, 05C48, 60B11, 28C20 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 $Θ(|V_n|^{\frac23})$ 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. |
| title | Critical level-set percolation on finite graphs and spectral gap |
| topic | Probability Combinatorics 60K35, 82B43, 60G60, 05C80, 05C48, 60B11, 28C20 |
| url | https://arxiv.org/abs/2601.07802 |