Geometry of Gaussian free field sign clusters and random interlacements

Abstract

For a large class of amenable transient weighted graphs G, we prove that the sign clusters of the Gaussian free field on G fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices such as $${\mathbb {Z}}^d$$ Z d , for $$d\ge 3$$ d ≥ 3 , but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman (Ann Math 171(3):2039–2087, 2010), and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from Sznitman (Invent Math 187(3):645–706, 2012).

A characterization of strong percolation via disconnection

“…= 𝑣 ′ −𝑣 3 ∧ 𝑢−𝑣 ′ 3 . We will apply the results of [10] to the graph ℤ 𝑑 (with unit weights); see also the proof of [20,Lemma 5.16] for a similar argument in the context of the Gaussian free field. Let 𝐿 0 = 𝑀(𝑁 0 ) for a large positive integer parameter 𝑁 0 to be determined.…”

“…Note that the condition 𝑣 ′ − 𝑣 ⩽ 𝑐 (which ensures that the requirement '𝑢 ⩾ 2𝛿' in (5.2) holds) is harmless; indeed if 𝑣 is decreased in (5.3) then the bound continues to hold by monotonicity. In particular, by choosing first 𝛼 = 𝛼(𝑑) small enough and then 𝐿 0 = 𝐿 0 (𝑣, 𝑣 ′ , 𝛾 𝑀 ) sufficiently large, one can ensure that [10,Conditions (7.5) where the events ÛC 𝜀 𝑥,𝑛 refers to those defined in (5.4), but with parameters (𝑣 ′ , 𝑣) in place of (𝑣 ′ − 𝜀, 𝑣 + 𝜀). The presence of two opposite directions of monotonicity requires a small extension over the decoupling inequality stated in [10,Theorem 2.4]; alternatively one can resort to the stronger result (2.23) (extending Proposition 2.3) applied with 𝑘 = 2.…”

“…In particular, by choosing first 𝛼 = 𝛼(𝑑) small enough and then 𝐿 0 = 𝐿 0 (𝑣, 𝑣 ′ , 𝛾 𝑀 ) sufficiently large, one can ensure that [10,Conditions (7.5) where the events ÛC 𝜀 𝑥,𝑛 refers to those defined in (5.4), but with parameters (𝑣 ′ , 𝑣) in place of (𝑣 ′ − 𝜀, 𝑣 + 𝜀). The presence of two opposite directions of monotonicity requires a small extension over the decoupling inequality stated in [10,Theorem 2.4]; alternatively one can resort to the stronger result (2.23) (extending Proposition 2.3) applied with 𝑘 = 2. It is worth emphasizing however that the decoupling only needs to operate between boxes of a given size, 𝐿 𝑛 , separated by a large multiple 𝑐𝓁 0 𝐿 𝑛 of their diameter, as originally in [38].…”

“…In order to conclude, one applies [10,Lemma 8.6], which implies that whenever ÛC 𝜀 0,𝑛 occurs, any two connected sets in 𝐵 𝜆𝐿 𝑛 , each of diameter at least (𝜆∕20)𝐿 𝑛 , are connected by a path 𝛾 ⊂ 𝐵 2𝜆𝐿 𝑛 such that ÛC 𝜀 𝑥,0 occurs for all 𝑥 ∈ 𝛾. This is readily seen to imply UC(𝑟, 𝑣 ′ , 𝑣) for 𝑟 = 𝑟 𝑛 = ⌊(𝜆∕10)𝐿 𝑛 ⌋, hence (5.5) yields (5.3) for such 𝑟.…”