Incipient infinite clusters and volume growth for Gaussian free fields and loop soups on metric graphs

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).