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Upper bounds on the one-arm exponent for dependent percolation models

Abstract

We prove upper bounds on the one-arm exponent \(\eta _1\) for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson–Voronoi and Poisson–Boolean percolation. More precisely, in dimension \(d=2\) we prove that \(\eta _1 \le 1/3\) for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann–Fock field), and in \(d \ge 3\) we prove \(\eta _1 \le d/3\) for finite-range fields, both discrete and continuous, and \(\eta _1 \le d-2\) for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.

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Notes

  1. Although in [46] this is proven only for finite Gaussian vectors one can deduce positive associations for all increasing events considered in this paper via standard approximation arguments, see [48, Lemma A.12].

  2. The argument is attributed to van den Berg.

  3. Although this lemma is stated for differentiable functions, it is easy to check that the proof goes through without differentiability since it only uses \(f(b) - f(a) \ge \int _a^b \frac{d^-}{d x} f(x) dx\).

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Acknowledgements

The second author was partially supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467. The authors thank Damien Gayet, Tom Hutchcroft, Ioan Manolescu and Hugo Vanneuville for helpful discussions, comments on an earlier draft, and for pointing out references [17] (Ioan) and [9, 55] (Hugo), and an anonymous referee for valuable comments that helped improve the presentation of the paper. This work was initiated while the first author was visiting the second author at Queen Mary University of London and we thank the University for its hospitality.

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The second author was partially supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467.

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Appendix A: Orthogonal decomposition of \(f_S\)

Appendix A: Orthogonal decomposition of \(f_S\)

For completeness we present a classical orthogonal decomposition of the Gaussian field on \(\mathbb {X}^d\), \(\mathbb {X}\in \{\mathbb {R}, \mathbb {Z}\}\),

$$\begin{aligned} f_S(\cdot ) = (q \star W|_S)(\cdot ) = \int _{y \in S} q(\cdot -y) dW(y), \end{aligned}$$

where \(S \subset \mathbb {T}^d\) is a compact domain, \(q \in L^2(\mathbb {T}^d)\), and W is the white noise on \(\mathbb {T}^d\). In the case \(\mathbb {X}= \mathbb {R}\) we also assume that \(f_S\) is continuous.

Proposition A.1

(Orthogonal decomposition of \(f_S\)) Let \((Z_i)_{i \ge 1}\) be a sequence of i.i.d. standard normal random variables and let \((\varphi _i)_{i \ge 1}\) be an orthonormal basis of \(L^2(S)\). Then, as \(n \rightarrow \infty \),

$$\begin{aligned} f^n_S:= \sum _{i \ge 1}^n Z_i (q \star \varphi _i) \Rightarrow f_S \end{aligned}$$

in law with respect to the \(C^0(\mathbb {X}^d)\)-topology on compact sets. In particular,

$$\begin{aligned} f_S(\cdot ) {\mathop {=}\limits ^{d}} \frac{Z_1 (q \star \mathbbm {1}_S)(\cdot )}{\sqrt{\text {Vol}(S)}} + g(\cdot ) \end{aligned}$$

where g is a Gaussian field independent of \(Z_1\).

Proof

Consider the case \(\mathbb {X}=\mathbb {R}\). Remark that, for each \(x \in \mathbb {R}^d\), \(f^n_S(x) \Rightarrow f_S(x)\) in law since they are centred Gaussian random variables and

$$\begin{aligned} \mathbb {E} \left[ \left( \sum _{i \ge 1}^n Z_i (q \star \varphi _i)(x) \right) ^2 \right]= & {} \sum ^n_{i \ge 1} \left( \int _S q(x-s) \varphi _i(s) \, dx \right) ^2 \rightarrow \int _S q(x-s)^2 \, dx \\= & {} \mathbb {E}[f_S(x)^2] \end{aligned}$$

by Parseval’s identity. Note also that the functions \(q \star \varphi _i\) are continuous (as a convolution of \(L^2\) functions), and so each \(f^n_S\) is continuous. Hence the first statement of the proposition follows by an application of Lemma A.2 below. For the second statement, set \(\varphi _1\) to be constant on S.

The case \(\mathbb {X}= \mathbb {Z}\) is similar but simpler; in fact \(L^2(S)\) is finite-dimensional in that case, so \(f_S^n {\mathop {=}\limits ^{d}} f_S\) for sufficiently large n. \(\square \)

Lemma A.2

Let \((f_i)_{i \ge 1}\) be a sequence of independent continuous centred Gaussian fields on \(\mathbb {R}^d\) and define \(g_n:=\sum ^n_{i \ge 1} f_i\). Suppose there exists a continuous Gaussian field g on \(\mathbb {R}^d\) such that, for each \(x \in \mathbb {R}^d\), \(g_n(x) \Rightarrow g(x)\) in law. Then \(g_n \Rightarrow g\) in law with respect to the \(C^0\)-topology on compact sets.

Proof

We follow the proof of [1, Theorem 3.1.2]. Since \(g_n(x)\) is a sum of independent random variables converging in law, by Levy’s equivalence theorem we may define g(x) as the almost sure limit of \(g_n(x)\). Fix a compact set \(\Omega \subset \mathbb {R}^d\), and consider \((g_n)_{n \ge 1}\) as elements of the Banach space \(C(\Omega )\) of continuous functions on \(\Omega \) equipped with the \(C_0\)-topology. By the Itô-Nisio theorem [1, Theorem 3.1.3], it suffices to show that

$$\begin{aligned} \int _\Omega g_n \, d \mu \rightarrow \int _\Omega g \, d \mu \end{aligned}$$

in mean (and so in probability) for every finite signed Borel measure \(\mu \) on \(\Omega \). Define the continuous functions \(u_n(x):= \mathbb {E}[g_n(x)^2]\) and \(u(x):= \mathbb {E}[g(x)^2]\). Then

$$\begin{aligned}{} & {} \mathbb {E}\left[ \left| \int _\Omega g \, d \mu - \int _\Omega g_n \, d \mu \right| \right] \le \int _\Omega \left( \mathbb {E}\left[ \left( g(x) - g_n(x) \right) ^2 \right] \right) ^{1/2} \, |\mu |(dx)\nonumber \\{} & {} \quad \le \int _\Omega \left( u(x) - u_n(x) \right) ^{1/2} |\mu |(dx). \end{aligned}$$

Since \(u_n \rightarrow u\) monotonically, by Dini’s theorem the convergence is uniform on \(\Omega \), so we have that \( \mathbb {E}[ | \int _\Omega g \, d \mu - \int _\Omega g_n \, d \mu | ] \rightarrow 0\) as required. \(\square \)

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Dewan, V., Muirhead, S. Upper bounds on the one-arm exponent for dependent percolation models. Probab. Theory Relat. Fields 185, 41–88 (2023). https://doi.org/10.1007/s00440-022-01176-3

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Keywords

  • Percolation
  • Critical exponents
  • Gaussian fields

Mathematics Subject Classification

  • Primary 60G60
  • Secondary 60F99