Incipient infinite clusters and volume growth for Gaussian free fields and loop soups on metric graphs (2412.05709v2)

Abstract: In this paper, we establish the existence and equivalence of four types of incipient infinite clusters (IICs) for the critical Gaussian free field (GFF) level-set and the critical loop soup on the metric graph $\widetilde{\mathbb{Z}}^d$ for all $d\ge 3$ except the critical dimension $d=6$. These IICs are defined as four limiting conditional probabilities, involving different conditionings and various ways of taking limits: (1) conditioned on ${0 \leftrightarrow{} \partial B(N)}$ at criticality (where $0$ is the origin of $\mathbb{Z}^d$, and $\partial B(N)$ is the boundary of the box $B(N)$ centered at $0$ with side length $2N$), and letting $N\to \infty$; (2) conditioned on ${0\leftrightarrow{} \infty}$ at super-criticality, and letting the parameter tend to the critical threshold; (3) conditioned on ${0 \leftrightarrow{} x}$ at criticality (where $x\in \mathbb{Z}^d$ is a lattice point), and letting $x\to \infty$; (4) conditioned on the event that the capacity of the critical cluster containing $0$ exceeds $T$, and letting $T\to \infty$. Our proof employs a robust framework of Basu and Sapozhinikov (2017) for constructing IICs as in (1) and (2) for Bernoulli percolation in low dimensions (i.e., $3\le d\le 5$), where a key hypothesis on the quasi-multiplicativity is proved in our companion paper. We further show that conditioned on ${0 \leftrightarrow{} \partial B(N)}$, the volume of the critical cluster containing $0$ within $B(M)$ is typically of order $M^{(\frac{d}{2}+1)\land 4}$, as long as $N\gg M$. This phenomenon indicates that the critical cluster of the GFF or the loop soup exhibits self-similarity, which supports Werner's conjecture (2016) that such cluster has a scaling limit. Moreover, the exponent of $M^{(\frac{d}{2}+1)\land 4}$ matches the conjectured fractal dimension of the scaling limit proposed by Werner (2016).