2024/12/07 by Zhenhao Cai, Jian Ding, Cai, Zhenhao +1
Computer Science · Mathematics · #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.05709
openalex publication_date 2024/12/07 · openalex created_date 2024/12/12 · openalex updated_date 2026/07/28
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 \widetildeℤd for all d≥ 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 ↔ ∂ B(N)\ at criticality (where 0 is the origin of ℤd, and ∂ B(N) is the boundary of the box B(N) centered at 0 with side length 2N), and letting N→ ∞; (2) conditioned on \0↔ ∞\ at super-criticality, and letting the parameter tend to the critical threshold; (3) conditioned on \0 ↔ x\ at criticality (where x∈ ℤd is a lattice point), and letting x→ ∞; (4) conditioned on the event that the capacity of the critical cluster containing 0 exceeds T, and letting T→ ∞. 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≤ d≤ 5), where a key hypothesis on the quasi-multiplicativity is proved in our companion paper. We further show that conditioned on \0 ↔ ∂ B(N)\, the volume of the critical cluster containing 0 within B(M) is typically of order M((d)/(2)+1)∧ 4, as long as N≫ 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((d)/(2)+1)∧ 4 matches the conjectured fractal dimension of the scaling limit proposed by Werner (2016).