Anomalous scaling law for the two-dimensional Gaussian free field
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arXiv
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| Format: | Preprint |
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2025
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| author | Rodriguez, Pierre-François Zhang, Wen |
| author_facet | Rodriguez, Pierre-François Zhang, Wen |
| contents | We consider the Gaussian free field $φ$ on $\mathbb{Z}^2$ at large spatial scales $N$ and give sharp bounds on the probability $θ(a,N)$ that the radius of a finite cluster in the excursion set $\{φ\geq a\}$ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which $θ(a,N)$ transitions from fractional logarithmic decay for near-critical parameters $(a,N)$ to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale $ξ$, which is such that $θ(a,N) \sim θ(0,ξ)(\tfrac{N}ξ)^{-τ}$ for typical heights $a$ as $N/ξ$ diverges, with an explicit exponent $τ$ that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in ${N}/ξ$ away from it. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anomalous scaling law for the two-dimensional Gaussian free field Rodriguez, Pierre-François Zhang, Wen Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 We consider the Gaussian free field $φ$ on $\mathbb{Z}^2$ at large spatial scales $N$ and give sharp bounds on the probability $θ(a,N)$ that the radius of a finite cluster in the excursion set $\{φ\geq a\}$ on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which $θ(a,N)$ transitions from fractional logarithmic decay for near-critical parameters $(a,N)$ to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale $ξ$, which is such that $θ(a,N) \sim θ(0,ξ)(\tfrac{N}ξ)^{-τ}$ for typical heights $a$ as $N/ξ$ diverges, with an explicit exponent $τ$ that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in ${N}/ξ$ away from it. |
| title | Anomalous scaling law for the two-dimensional Gaussian free field |
| topic | Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 |
| url | https://arxiv.org/abs/2512.10933 |