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Liouville dynamical percolation

Abstract

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a \(\gamma \)-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the \(\gamma \)-Liouville measure associated with a two-dimensional log-correlated field h. Roughly speaking, this continuum percolation process evolves very rapidly where the field h is high and barely moves where the field h is low. Our main results can be summarized as follows.

  • First, we build this inhomogeneous dynamical percolation, which we call \(\gamma \)-Liouville dynamical percolation (LDP), by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: \(\gamma \in [0,2-\sqrt{5/2})\), \(\gamma \in [2-\sqrt{5/2}, \sqrt{3/2})\), and \(\gamma \in (\sqrt{3/2},2)\).

  • When \(\gamma <\sqrt{3/2}\), we prove that \(\gamma \)-LDP is mixing in the Schramm–Smirnov space as \(t\rightarrow \infty \), quenched in the log-correlated field h. On the contrary, when \(\gamma >\sqrt{3/2}\) the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for \(\gamma =\sqrt{1/6}\) is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations.

  • When \(\gamma <\sqrt{3/4}\), we obtain quantitative bounds on the mixing of quad crossing events.

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Fig. 1

Notes

  1. The notion “Liouville dynamical percolation” (LDP) may refer to either dLDP or cLDP, and the meaning will be clear from the context.

  2. See Sect. 2.4 for the precise class of fields we consider.

  3. We remark that the loop ensemble space considered in e.g. [8] and the quad crossing space considered in [26, 57] and this paper are equivalent in the sense that the associated \(\sigma \)-algebras are the same. See [8] and [24, Section 2.3] for a proof that the loops determine the quad crossing information, and see [33, Theorem 6.10] for the converse result. Therefore \((\omega ^\gamma (t))_{t\ge 0}\) can also be viewed as a process with values in the loop ensemble space.

  4. Above we defined \(B^{{\text {h}}}_r(x)\) only for \(x\in {\mathbb {T}}_{\eta }\), but the definition extends immediately to \(x\in {\mathbb {C}}\) by considering \({\mathbb {T}}_\eta \) recentred so that \(x\in {\mathbb {T}}_\eta \).

  5. We will choose \(\varrho \) such that Lemma 2.14 is satisfied.

  6. A concrete way to understand the limiting pivotal points is through the limiting loop ensemble called \({{\,\mathrm{CLE}\,}}_6\) [8]. The limiting pivotal points can be realized as the collection of double points and intersection points of loops in \({{\,\mathrm{CLE}\,}}_6\). Since our paper is mainly focused on the Schramm–Smirnov topology, we will not elaborate on this point of view. See [33, Section 5] for more details.

  7. In [23], the distinction is made between the spectral sample \({\mathscr {S}}_\eta \) viewed as a random set and the counting measure \(\lambda _\eta \) on the spectral sample \({\mathscr {S}}_\eta \). Here we identify the concepts for convenience.

  8. We identify functions that differ by a constant since \((f,g)=0\) if \(f\equiv a\) or \(g\equiv a\) for some constant a.

  9. One can check that, in the notation of [26], both \(\boxminus _Q^c\) and \(\boxdot _U^c\) are open for the topology generated by \(d_{\mathscr {H}}^{{\text {mod}}}\). Furthermore, it is possible to see that this topology is also finer than the one presented in [26, Section 2.3] (as long as \(\emptyset \) is considered to be a quad).

  10. For example, consider the sequence of elements in \({\mathscr {H}}\) such that the nth element consists of the quad \(Q_n( (x,y) )=((1-1/n)x,y)\) and all quads Q satisfying \(Q\le Q_n\). This sequence is Cauchy, but does not have a limit, since (by the requirement that the elements of \({\mathscr {H}}\) are closed) the limiting object would need to contain the quad \(Q(z)\equiv z\), while the limit cannot contain this quad by definition of \(d_{\mathscr {H}}^{{\text {mod}}}\).

  11. In case z lies on the grid (i.e., at least one of its coordinates is an integer multiple of \(\epsilon \)), choose \(A_1\) arbitrarily among the possible squares.

  12. Notice that \(\mathcal Q^o\) is always non-empty: There exist configurations where all the quads are crossed (e.g. if all sites of \({\mathbb {T}}\) are open) and there exist configurations where the quads are not all crossed (e.g. if all sites of \({\mathbb {T}}\) are closed). By moving from one configuration to the other by changing the sites one by one, we see that there exist configurations where we have a pivotal point.

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Acknowledgements

We thank Jeffrey Steif for a very good comment and discussion after our talk in Oberwolfach, which lead to Sect. 2.6. We also thank Rick Bradley and Jean-Paul Thouvenot for their inputs to that section, we thank Ewain Gwynne for helpful comments to the paper, and we thank the anonymous referee for many helpful comments and careful reading of the paper. The research of C.G. is supported by the ERC grant LiKo 676999 and Institut Universitaire de France (IUF). The research of N.H. is supported by Dr. Max Rössler, the Walter Haefner Foundation, the ETH Zürich Foundation, and a fellowship from the Norwegian Research Council. The research of A.S was supported by the ERC grant LiKo 676999 and is now supported by Grant ANID AFB170001 and FONDECYT iniciación de investigación No 11200085. The research of X.S. was supported by Simons Society of Fellows under Award 527901, and by NSF Award DMS-1811092 and DMS-2027986. Part of the work on this paper was carried out during the visit of N.H. and X.S. to Lyon in November 2017 and 2018. They thank for the hospitality and for the funding through the ERC grant LiKo 676999. A.S would also like to thank the hospitality of Núcleo Milenio “Stochastic models of complex and disordered systems” for repeated invitation to Santiago, where part of this paper was written.

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Appendices

Appendix

Limit properties of LQG measures

1.1 Continuity of LQG measures

Let \(\gamma \ge 0\), let \(\sigma ^{n}\) be a sequence of random measures in a bounded domain \(D\subset {\mathbb {C}}\) converging in probability for the Prokhorov topology to a measure \(\sigma \) with finite total mass, and let \(\mu ^{n}_{\gamma h}=\mu ^{\sigma ^n}_{\gamma h}\) be the sequence of \(\gamma \)-LQG measures of h with respect to \(\sigma _n\). The goal of this section is to give a sufficient condition for \(\mu ^{n}_{\gamma h}\) to converge to \(\mu _{\gamma h}^{\sigma }\), the \(\gamma \)-LQG of h with respect to \(\sigma \).

We will use several estimates from [5], where it was proved that \(\mu ^\sigma _{\gamma h}\) is the limit of \(\mu ^\sigma _{\gamma h_\epsilon }\) in \(L^1\) for \(h_\epsilon \) a smooth approximation to h (see (2.12)). Many notations will be borrowed from that paper, and it is advisable that the reader is familiar with that paper before reading the proof.

Proposition A.1

Take \(d >0\) and assume that \(\sup _n {\mathbb {E}}[{\mathcal {E}}_{d}(\sigma ^{n})]<\infty \). Then, for all \(\gamma ^2<2d\) and deterministic sets \({\mathcal {O}}\subseteq {\mathbb {C}}\) such that \(\sigma (\partial {\mathcal {O}})=0\) a.s., the LQG measures considered above are well-defined and we have that \(\mu ^{n}_{\gamma h}({\mathcal {O}})\rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {O}})\) in \(L^1\). Furthermore, \(\mu ^{n}_{\gamma h}|_{{\mathcal {O}}}\) converges in probability to \(\mu _{\gamma h}^{\sigma }|_{{\mathcal {O}}}\) in the topology of weak convergence of measures on \({\mathcal {O}}\). If \(\gamma ^2<d\) and \(\sigma ^n\rightarrow \sigma \) in \(L^2\), then \(\mu ^{n}_{\gamma h}({\mathcal {O}})\rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {O}})\) in \(L^2\).

Proof

Fix \(\gamma ^2<2d\). For simplicity we write \(\mu ^{n}_{\gamma h}\) as \(\mu ^{n}\). For some smooth approximation \(h_\epsilon \) to h (e.g. the circle average approximation) we write \(\mu ^{n}_{\gamma h_\epsilon }\) as \(\mu ^{n}_\epsilon \). Similarly, we write \(\mu \) and \(\mu _\epsilon \) when the base measure is \(\sigma \) instead of \(\sigma ^n\). By the triangle inequality, for \(n\in {\mathbb {N}}\),

$$\begin{aligned} {\mathbb {E}}|\mu ^{n}({\mathcal {O}})- \mu ({\mathcal {O}})| \le {\mathbb {E}}|\mu ^{n}({\mathcal {O}})- \mu ^n_{\epsilon }({\mathcal {O}})| + {\mathbb {E}}|\mu ^{n}_{\epsilon }({\mathcal {O}})- \mu _{\epsilon }({\mathcal {O}})| + {\mathbb {E}}|\mu _{\epsilon }({\mathcal {O}})- \mu ({\mathcal {O}})|\,. \end{aligned}$$

Since \(h_\epsilon \) is smooth, \(\sigma (\partial {\mathcal {O}})=0\), and \(\sigma ^n\rightarrow \sigma \) in probability, we get that \(\mu ^{n}_{\epsilon }({\mathcal {O}})\rightarrow \mu _{\epsilon }({\mathcal {O}})\) in probability. Using \(\sup _n {\mathbb {E}}[{\mathcal {E}}_{d}(\sigma ^{n})]<\infty \), this gives that the second term on the right side converges to 0 as \(n\rightarrow \infty \) for any fixed \(\epsilon \). The third term converges to 0 as \(\epsilon \rightarrow 0\) by e.g. the main result of [5]. Therefore, to show that \(\mu ^{n}_{\gamma h}({\mathcal {O}})\rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {O}})\) in \(L^1\) it is sufficient to handle the first term, i.e., to show that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\sup _{n\in {\mathbb {N}}} {\mathbb {E}}\left[ |\mu ^{n}({\mathcal {O}})-\mu ^{n}_\epsilon ({\mathcal {O}})| \right] =0. \end{aligned}$$

This result follows from a close inspection of [5].

For some \(\epsilon _0\le 1\) to be determined right below, define the following event \(G_\epsilon ^\alpha (x)\), which, roughly speaking, says that the field h is not too large close to x:

$$\begin{aligned} G_\epsilon ^\alpha (x) = \{ h_r(x) \le \alpha \log (1/r) \text { for all } r\in [\epsilon ,\epsilon _0] \}\,. \end{aligned}$$

Then define

$$\begin{aligned} I_{\epsilon }^n:=\int _{{\mathcal {O}}} {\mathbf {1}}_{(G_\epsilon ^a)^c(x)} \mu ^n_\epsilon (d^2x), \ \ \ J_\epsilon ^n:=\int _{{\mathcal {O}}} {\mathbf {1}}_{G_\epsilon ^a(x)} \mu ^n_\epsilon (d^2x), \end{aligned}$$

and note that \(\mu _\epsilon ^n({\mathcal {O}})=I_\epsilon ^n+J_\epsilon ^n\). By [5, Lemma 3.2], for all \(\eta >0\) there exists \(\epsilon _0>0\) such that \(\sup _{n\in {\mathbb {N}}} {\mathbb {E}}\left[ I_\epsilon ^n\right] \le \eta \) for all \(\epsilon \in (0,\epsilon _0)\); we fix \(\epsilon _0>0\) such that this condition is satisfied. It is sufficient to show the following:

$$\begin{aligned} \lim _{\epsilon ,\epsilon '\rightarrow 0}\sup _{n\in {\mathbb {N}}}{\mathbb {E}}[(J_\epsilon ^n-J_{\epsilon '}^n)^2]=0\,. \end{aligned}$$

We will prove this by showing the existence of a function \(F:{\mathcal {O}}\times {\mathcal {O}}\rightarrow {\mathbb {R}}\) such that uniformly in n,

$$\begin{aligned} {\mathbb {E}}\left[ (J_\epsilon ^n)^2\right] ,\,{\mathbb {E}}\left[ J_\epsilon ^n J_{\epsilon '}^n\right] \rightarrow \iint _{{\mathcal {O}}\times {\mathcal {O}}} F(x,y)\,\sigma ^n(d^2x)\sigma ^n(d^2y) \quad \text { as }\epsilon ,\epsilon '\rightarrow 0. \end{aligned}$$

We just treat \({\mathbb {E}}\left[ (J_\epsilon ^n)^2\right] \), since \({\mathbb {E}}\left[ J_\epsilon ^n J_{\epsilon '}^n\right] \) is treated in the same way. We have

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ (J_\epsilon ^n)^2\right]&=\iint _{|x-y|\le \delta } e^{\gamma ^2{\mathbb {E}}\left[ h_\epsilon (x)h_\epsilon (y)\right] }\widetilde{\mathbb {P}}(G_\epsilon (x)\cap G_\epsilon (y))\,\sigma ^n(d^2x)\sigma ^n(d^2y)\\&\quad + \iint _{|x-y|\ge \delta } e^{\gamma ^2{\mathbb {E}}\left[ h_\epsilon (x)h_\epsilon (y)\right] }\widetilde{\mathbb {P}}(G_\epsilon (x)\cap G_\epsilon (y))\,\sigma ^n(d^2x)\sigma ^n(d^2y), \end{aligned}\nonumber \\ \end{aligned}$$
(A.1)

where \(\widetilde{\mathbb {P}}\) is a certain probability measure absolutely continuous with respect to \({\mathbb {P}}\) (defined above [5, equation (3.8)]). Now [5, equation (3.12)] shows that we can find \(\beta <d\) (which corresponds to choosing a nice \(\alpha >0\) in [5]) such that for all \(n\in {\mathbb {N}}\), the first term on the right side of (A.1) is smaller than a constant depending only on the correlation kernel of h times

$$\begin{aligned} \iint _{|x-y|\le \delta } |x-y|^{-\beta } \sigma ^n(d^2x)\sigma ^n(d^2y)\le \delta ^{\overline{\beta }-\beta } \sup _{n} {\mathcal {E}}_{\overline{\beta }}(\sigma ^n), \end{aligned}$$

where \(\overline{\beta }\) is chosen such that \(\overline{\beta }\in (\beta ,d)\). Given \(\eta >0\), let us choose \(\delta >0\) such that the first term on the right side of (A.1) is smaller than \(\eta \). Now, as in [5, Lemma 4.1], when \(|x-y|\ge \delta \), we have that \(e^{\gamma ^2{\mathbb {E}}\left[ h_\epsilon (x)h_\epsilon (y)\right] }\widetilde{\mathbb {P}}(G_\epsilon (x)\cap G_\epsilon (y))\) converges in the topology of uniform convergence to a function F(xy). Thus, uniformly in n, the second term on the right side of (A.1) converges to \(\iint _{|x-y|\ge \delta }F(x,y)\sigma ^n(d^2x)\sigma ^n(d^2y)\). Since \(\eta \) was arbitrary, this concludes the proof that \(\mu ^{n}_{\gamma h}({\mathcal {O}})\rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {O}})\) in \(L^1\).

The next assertion of the lemma is that \(\mu ^{n}_{\gamma h}|_{{\mathcal {O}}}\) converges in probability to \(\mu _{\gamma h}^{\sigma }|_{{\mathcal {O}}}\) in the topology of weak convergence of measures on \({\mathcal {O}}\). The proof can be carried out exactly as in [5] and is therefore omitted.

To conclude the proof of the lemma, we will argue that we also have \(L^2\) convergence if \(\gamma ^2<d\) and \(\sigma ^n\rightarrow \sigma \) in \(L^2\). We have

$$\begin{aligned}&{\mathbb {E}}|\mu ^{n}({\mathcal {O}})- \mu ({\mathcal {O}})|^2 \\&\quad \le 3{\mathbb {E}}|\mu ^{n}({\mathcal {O}})- \mu ^n_{\epsilon }({\mathcal {O}})|^2 + 3{\mathbb {E}}|\mu ^{n}_{\epsilon }({\mathcal {O}})- \mu _{\epsilon }({\mathcal {O}})|^2 + 3{\mathbb {E}}|\mu _{\epsilon }({\mathcal {O}})- \mu ({\mathcal {O}})|^2\,. \end{aligned}$$

The second term on the right side converges to 0 since \(h_\epsilon \) is smooth and by the assumption that \(\sigma ^n\rightarrow \sigma \) in \(L^2\). The third term on the right side converges to 0 by e.g. [5]. The proof that the first term converges to 0 can be done as in the \(L^1\) case, except that we may choose \(\alpha >2\), which implies that \(G_\epsilon ^\alpha (x)\) does not occur for any x a.s., so \(I^n_\epsilon =0\) a.s. \(\square \)

It is possible to bound uniformly the expected energy of the measure \(\lambda ^\epsilon \) and its approximations.

Proposition A.2

For any \(\epsilon >0\) and \(d<3/4\),

$$\begin{aligned} \sup _{\eta \in (0,1]}{\mathbb {E}}\left[ {\mathcal {E}}_d(\lambda ^\epsilon (\omega _\eta )) \right] <\infty . \end{aligned}$$

Proof

This follows by the argument in the proof of [23, Lemma 4.5], where it is proved via quasi-multiplicativity that \({\mathbb {E}}[\lambda ^\epsilon (\omega _\eta )^2]<\infty \). \(\square \)

1.2 Convergence to 0 of Liouville measures

Let \(\beta >0\) and assume that \(\sigma _\eta \) is a sequence of measures that can be written as

$$\begin{aligned} \sigma _\eta (d^2z) = C_\eta \eta ^{-\beta } \sum _{x\in I_\eta \subseteq {\mathbb {T}}_\eta } {\mathbf {1}}_{z\in B^{{\text {h}}}_\eta (x)}\,d^2z, \end{aligned}$$
(A.2)

where \(C_\eta =\eta ^{o(1)}\) is a deterministic sequence and \(I_\eta \) is a (possibly random) set independent of h. Furthermore, assume that the expected total mass of the measure is bounded uniformly in \(\eta \), i.e.,

$$\begin{aligned} \sup _{\eta >0}C_\eta \eta ^{2-\beta }\sum _{z\in {\mathbb {T}}_\eta }{\mathbb {P}}[z\in I_\eta ]<\infty \,. \end{aligned}$$
(A.3)

We are interested in seeing when the Liouville measure associated to this measure converges to 0. In particular, we are interested in proving the following proposition.

Proposition A.3

Assume \(0\le 2-\beta <\gamma ^2/2\). Then the sequence of Liouville measures \(\mu _{\gamma h}^{\sigma _\eta }\) converges to 0 in probability as \(\eta \rightarrow 0\).

Proof

We just need to prove that \(\mu _{\gamma h}^{\sigma _\eta }(D)\rightarrow 0\) in probability. To do that, let us define

$$\begin{aligned} A_\eta :=\{z\in D: \mu _{\gamma h}(B^{{\text {h}}}_\eta (z))<\eta ^{2-\gamma ^2/2+\delta }\} \end{aligned}$$

for some \(\delta >0\) to be determined. We have

$$\begin{aligned} \mu _{\gamma h}^{\sigma _\eta }(D)= & {} C_\eta \eta ^{-\beta } \sum _{z\in I_\eta \subseteq {\mathbb {T}}_\eta } {\mathbf {1}}_{z\in A_\eta } \mu _{\gamma h}(B^{{\text {h}}}_\eta (z))\nonumber \\&+C_\eta \eta ^{-\beta } \sum _{z\in I_\eta \subseteq {\mathbb {T}}_\eta } {\mathbf {1}}_{z\notin A_\eta } \mu _{\gamma h}(B^{{\text {h}}}_\eta (z))\,. \end{aligned}$$
(A.4)

First we show that the first term on the right side of (A.4) converges to 0 in \(L^1\). To do that, let us recall [4, Proposition 4.1 and Corollary 6.2], which say that for any log-correlated field and any \(q<4/\gamma ^2\),

$$\begin{aligned} {\mathbb {E}}\left[ \mu _{\gamma h}(B^{{\text {h}}}_r(z))^q \right] \le r^{ -\gamma ^2 q^2/2 + (2+\gamma ^2/2)q +O(1)}, \end{aligned}$$
(A.5)

where the O(1) is uniform in z. Thus, for any \(p>0\) the expected value of the first term on the right side of (A.4) is upper bounded by

$$\begin{aligned}&{\mathbb {E}}\left[ C_\eta \eta ^{-\beta } \sum _{z\in I_\eta \subseteq {\mathbb {T}}_\eta } \mu _{\gamma h}(B^{{\text {h}}}_\eta (z))^{1-p} \eta ^{(2-\gamma ^2/2+\delta )p}\right] \\&\quad \le \eta ^{-\gamma ^2(1-p)^2/2+(2+\gamma ^2/2)(1-p)+o(1)}\eta ^{(2-\gamma ^2/2+\delta )p} \eta ^{-\beta }\sum _{z\in {\mathbb {T}}}{\mathbb {P}}[z\in I_{\eta }]\\&\quad =\eta ^{-\gamma ^2p^2/2+\delta p+O(1)}. \end{aligned}$$

Therefore, for any \(\delta >0\) we can find \(p>0\) sufficiently small such that the first term on the right side of (A.4) goes to 0 in probability.

Now, we will show that the second term on the right side of (A.4) converges to 0 in probability. By Markov’s inequality,

$$\begin{aligned} {\mathbb {P}}(\mu _{\gamma h}(B^{{\text {h}}}_\eta (z))>\eta ^{2-\gamma ^2/2+\delta }) \le \eta ^{ 2 - \left( 2+\delta -\frac{\gamma ^2}{2}\right) +O(1)} =\eta ^{\frac{\gamma ^2}{2}-\delta +O(1)}, \end{aligned}$$

where again, the O(1) can be taken uniformly for all \(z\in I_\eta \). Therefore we can upper bound the probability that the second term on the right side of (A.4) is bigger than 0 by

$$\begin{aligned} {\mathbb {P}}(I_\eta \backslash A_\eta \ne \emptyset )&\le {\mathbb {E}}\left[ {\mathbb {E}}\left[ \sum _{z\in I_\eta } {\mathbf {1}}_{A_{z,\eta }} \,\Big |\, I_\eta \right] \right] \le {\mathbb {E}}\left[ |I_\eta | \right] \sup _{z}{\mathbb {P}}(A_{z,\eta }^c) \\&\le \eta ^{-2+\beta }\eta ^{\frac{\gamma ^2}{2}-\delta +O(1)}{\mathop {\longrightarrow }\limits ^{\eta \rightarrow 0}} 0, \end{aligned}$$

where we have taken \(2\delta =\gamma ^2/2-2+\beta >0\). This is enough to conclude. \(\square \)

1.3 Convergence of the modified Liouville measure

As in the section before, we work with measures of the type (A.2). However, we now assume that \(\gamma \in (0,\sqrt{3/2})\). We add the assumption that \(\sigma _\eta \rightarrow \sigma \) a.s, and that \(\sup _\eta {\mathbb {E}}[{\mathcal {E}}_d(\sigma _\eta )]<\infty \) for a fixed \(d>\gamma ^2/2\). Let us note that Proposition A.1 implies that \(\mu _{\gamma h}^{\sigma _\eta }\rightarrow \mu _{\gamma h}^\sigma \) in probability for the weak topology as \(\eta \rightarrow 0\). The issue we address in this section is the convergence of the measure

$$\begin{aligned} \widetilde{\mu }_{\gamma h}^{C,\sigma _\eta }(d^2z):= C_\eta \eta ^{-\beta }\sum _{x\in I_\eta } {\mathbf {1}}_{z\in B^{{\text {h}}}_\eta (x)\cap {\mathcal {M}}_C} \mu _{\gamma h}(d^2z)\,. \end{aligned}$$

To do this, let us introduce the following set, where \(\varrho \) is as in (1.5),

$$\begin{aligned} {\mathcal {M}}^r_{C}:=\{x \in D: \mu _{\gamma h} (B^{{\text {h}}}_{2^{-n}}(x))<C\alpha ^{2^{-n}}_4(2^{-n},1)(2^{-n})^{\varrho }, \text { for all } n\le \lfloor \log _2(r)\rfloor \}, \end{aligned}$$

and show the following lemmas.

Lemma A.4

A.s. for any \(n\in {\mathbb {N}}\), the function \(x\mapsto \mu _{\gamma h} (B^{{\text {h}}}_{2^{-n}}(x))\) is continuous.

Proof

To see this let us define

$$\begin{aligned} f(r):=\sup _{x\in D}\mu _{\gamma h} (\partial B^{{\text {h}}}_r(x)\cap D). \end{aligned}$$

Note that the lemma follows from just showing that \({\mathbb {P}}(f(2^{-n})=0)=1\) for all \(n\in {\mathbb {N}}\).

First, let us see that f(r) is a measurable function of h. This follows because

$$\begin{aligned} f(r)= \inf _{\epsilon >0} \sup _{x\in {\mathbb {Q}}^2\cap D}\mu _{\gamma h} ( (B^{{\text {h}}}_{r+\epsilon }(x)\backslash B^{{\text {h}}}_{r-\epsilon }(x))\cap D). \end{aligned}$$

The edges of the hexagonal lattice dual to \({\mathbb {T}}_\eta \) have angle with the y-axis equal to 0, \(2\pi /3\), or \(4\pi /3\). Therefore, to conclude it is sufficient to show that a.s. no line in one of these three directions has positive mass. We will show this for lines parallel to the y-axis, but the two other directions can be treated by the exact same argument.

For simplicity we assume that \(D\subset [0,1]^2\); the exact same argument works for D contained in a larger square. For each \(n\in {\mathbb {N}}\) let \({\mathcal {I}}_n\) be a collection of \(2^n\) rectangles with disjoint interior contained in \([0,1]^2\) of the form \([k2^{-n},(k+1)2^{-n}]\times [0,1]\). By a union bound, in order to conclude it is sufficient to show that for any \(s>0\) and for all \(I\in {\mathcal {I}}_n\),

$$\begin{aligned} {\mathbb {P}}[ \mu _{\gamma h}(I)>s ] < o_n(1)2^{-n}s^{-1}, \end{aligned}$$
(A.6)

where the \(o_n(1)\) is uniform in I. Let \(\ell =\lceil 2^{n/2}\rceil \), and divide I into \(\ell \) disjoint rectangles of width \(2^n\) and height \(2^n/\ell \). Define a new log correlated field \(\widetilde{h}\) in \(\widetilde{D}_n=[0,\ell ]\times [0,2^n/\ell ]\) as follows. Divide \(\widetilde{D}_n\) into \(\ell \) disjoint rectangles of width \(2^n\) and height \(2^n/\ell \), and let \(\widetilde{\mathcal {I}}_n\) denote this collection of rectangles. For some arbitrary enumeration of \(\widetilde{\mathcal {I}}_n\) and \({\mathcal {I}}_n\) and \(j=1,\dots ,\ell \), set \(\widetilde{h}\) restricted to the jth rectangle of \(\widetilde{\mathcal {I}}_n\) equal to h restricted to the jth rectangle of \({\mathcal {I}}_n\). Let \({\check{h}}\) be equal to \(1+r\) times a log-correlated field in \([0,1]^2\) of the form (2.11) which is independent of n, where \(r>0\) is some small parameter to be determined; then the covariance kernel of \({\check{h}}\) is equal to \(-(1+r)^2\log |x-y|+(1+r)^2g(x,y)\). For sufficiently large n, the covariance kernel of \(\widetilde{h}\) will be pointwise smaller than the covariance kernel of \({\check{h}}|_{\widetilde{D}_n}\), so the by Kahane’s convexity inequality [35] (see also [4, 51]), we have \({\mathbb {E}}[ \mu _{\gamma h}(I)^{1+r} ]={\mathbb {E}}[ \mu _{\gamma \widetilde{h}}(\widetilde{D}_n)^{1+r} ]\le {\mathbb {E}}[ \mu _{\gamma {\check{h}}}(\widetilde{D}_n)^{1+r} ]\). Proceeding as in e.g. [4, Corollary 6.5] we have \({\mathbb {E}}[ \mu _{\gamma {\check{h}}}(\widetilde{D}_n)^{1+r}]\asymp 2^{n(1+r)/2}\) for r sufficiently small, so we get (A.6) by an application of Chebyshev’s inequality. \(\square \)

Lemma A.5

For any \(r>0\), we have that \(\mu _{\gamma h}^\sigma (\partial {\mathcal {M}}_C^r) =0\) a.s.

Proof

Let us define the set

$$\begin{aligned} E_{C}^n:=\{x \in D: \mu _{\gamma h} (B^{{\text {h}}}_{2^{-n}}(x))=C\alpha ^{2^{-n}}_4(2^{-n},1)2^{-n\varrho } \}, \end{aligned}$$

and note that Lemma A.4 implies that \(\mu _{\gamma h}^\sigma (\partial {\mathcal {M}}_C^r) \subseteq \bigcup E_{C}^n\). Thus, it is enough to show that \(\mu _{\gamma h}^\sigma (E_{C}^n)=0\). Thanks to Fubini’s theorem, it is sufficient to show that for any fixed \(x\in D\), a.s.,

$$\begin{aligned} {\mathbb {P}}[\mu _{\gamma h} (B^{{\text {h}}}_{2^{-n}}(x))=C\alpha ^{2^{-n}}_4(2^{-n},1)2^{-n\varrho }]=0. \end{aligned}$$
(A.7)

By the proof of [5, Lemma 5.1] we can write h on the form \(h=\alpha g+h'\), where g is a deterministic continuous function, \(\alpha \) is a standard normal random variable, and \(h'\) is a random log-correlated field independent of \(\alpha \). We may assume that g is not identically equal to zero in \(B^{{\text {h}}}_{2^{-n}}(x)\). Condition on \(h'\) and define the following random function

$$\begin{aligned} G_{h'}(a) = \mu _{\gamma (ag+h')} (B^{{\text {h}}}_{2^{-n}}(x)) =\int _{B^{{\text {h}}}_{2^{-n}}(x)} e^{\gamma a g(z)} d\mu _{\gamma h'}(z). \end{aligned}$$

By expanding \(e^{\gamma a g(z)}\) pointwise as a power series in a, we get that, conditioned on \(h'\), the function \(a\mapsto G_{h'}(a)\) is real analytic. By calculating the second derivative of \(a\mapsto G_{h'}(a)\), we see that the function is not constant. For any constant c, the set of points at which a non-constant analytic function is equal to c cannot have any accumulation points; otherwise all derivatives of the function would be zero at this accumulation point, and the function would be constant. In particular, the set of points at which the function is equal to c has zero Lebesgue measure. Since \(G_{h'}(\alpha )\overset{d}{=}\mu _{\gamma h} (B^{{\text {h}}}_{2^{-n}}(x))\) and \(\alpha \) is a standard normal independent of \(h'\), this implies (A.7). \(\square \)

Let us use this lemma to prove the following proposition.

Proposition A.6

For all \(\gamma \in (0,\sqrt{3/2})\) and all open \({\mathcal {O}}\subset D\) such that \(\sigma ({\mathcal {O}})<\infty \) and \(\sigma (\partial {\mathcal {O}})=0\) a.s., we have that \(\widetilde{\mu }_{\gamma h}^{C,\sigma _\eta }({\mathcal {O}}) =\mu _{\gamma h}^{\sigma _\eta }({\mathcal {O}}\cap {\mathcal {M}}_C) \rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {O}}\cap {\mathcal {M}}_C) =\widetilde{\mu }_{\gamma h}^{C,\sigma }({\mathcal {O}})\) in \(L^1\) as \(\eta \rightarrow 0\). Furthermore \(\widetilde{\mu }^{C,\sigma _\eta }_{\gamma h}|_{{\mathcal {O}}}\) converges in probability to \(\widetilde{\mu }_{\gamma h}^{C,\sigma }|_{{\mathcal {O}}}\) in the topology of weak convergence of measures on \({\mathcal {O}}\).

Proof

Let us start by fixing \(r>0\) and upper bounding \(| \mu _{\gamma h}^{\sigma }({\mathcal {M}}_C\cap {\mathcal {O}})-\widetilde{\mu }_{\gamma h}^{C,\sigma _\eta }({\mathcal {O}})|\) by

$$\begin{aligned} \begin{aligned}&|\mu _{\gamma h}^{\sigma }({\mathcal {M}}_C\cap {\mathcal {O}}) -\mu _{\gamma h}^{\sigma }({\mathcal {M}}_C^r\cap {\mathcal {O}})| + | \mu _{\gamma h}^{\sigma }({\mathcal {M}}_C^r\cap {\mathcal {O}})-\mu _{\gamma h}^{\sigma _\eta }({\mathcal {M}}_C^r\cap {\mathcal {O}})|\\&\quad + |\mu _{\gamma h}^{\sigma _\eta }({\mathcal {M}}_C^r\cap {\mathcal {O}}) -\widetilde{\mu }_{\gamma h}^{C,\sigma _\eta }({\mathcal {O}})|\,. \end{aligned} \end{aligned}$$
(A.8)

Let us first note that as \(r\rightarrow 0\) the first term converges to 0 a.s. thanks to the fact that \({\mathcal {M}}_C^r\downarrow {\mathcal {M}}_C\) as \(r\downarrow 0\). For the second term, we need to show is that for fixed \(r>0\), \(\widetilde{\mu }_{\gamma h}^{C,\sigma _\eta }( {\mathcal {M}}_C^r\cap {\mathcal {O}})=\mu _{\gamma h}^{\sigma _\eta }({\mathcal {M}}_C^r\cap {\mathcal {O}})\rightarrow \mu _{\gamma h}^{\sigma }({\mathcal {M}}_C^r\cap {\mathcal {O}})\) as \(\eta \rightarrow 0\). This is true thanks to the last assertion of Proposition A.1 and Lemma A.5. For the last term, we use that \({\mathcal {M}}_C^r\) is decreasing in r to show that it is equal to

$$\begin{aligned} C_\eta \eta ^{-\beta }\sum _{x\in I_\eta }\mu _{\gamma h}(B^{{\text {h}}}_\eta (x)\cap ({\mathcal {M}}_C^r\backslash {\mathcal {M}}_C)\cap {\mathcal {O}})\,. \end{aligned}$$

Thus, its expected value is bounded by a constant times

$$\begin{aligned} {\mathbb {E}}\left[ \mu _{\gamma h}({\mathcal {M}}_C^r\backslash {\mathcal {M}}_C)\right] \,. \end{aligned}$$
(A.9)

Note that this term is independent of \(\eta \), and that \({\mathcal {M}}_C^r\downarrow {\mathcal {M}}_C\) as \(r\downarrow 0\). Thus, we can use dominated convergence to show that this term converges to 0 uniformly in \(\eta \).

The last assertion is proved similarly as the last assertion of Proposition A.1. \(\square \)

Size of the spectral sample for multiple quad crossings

Let \(\mathcal Q\) be a collection of finitely many quads. For \(R>1\) let \(R\mathcal Q\) denote the same set of quads rescaled by R, i.e.,

$$\begin{aligned} R\mathcal Q:= \{RQ\,:\,Q\in \mathcal Q\}. \end{aligned}$$

Let \({\mathbb {T}}\) denote the triangular lattice where adjacent vertices have distance 1. For an instance \(\omega \) of critical site percolation on \({\mathbb {T}}\) let \(f(\omega )=f_{R\mathcal Q}(\omega )\) be the indicator function describing whether all the quads of \(R\mathcal Q\) have an open crossing. Throughout this section we do not rescale the triangular lattice; to be consistent with [23] we instead rescale the quads by R. Several observables throughout the section will depend on R, and by simplicity we will often omit the R dependence in notations.

For any set \(V\subset {\mathbb {C}}\) let \({\mathcal {A}}_\square (V,\mathcal Q)\) denote the event that the vertices in \(V\cap {\mathbb {T}}\) are pivotal for \(\mathcal Q\), i.e., there exists a percolation configuration \(\omega '\) such that \(\omega |_{{\mathbb {T}}\setminus V}=\omega '|_{{\mathbb {T}}\setminus V}\) and \(f(\omega )\ne f(\omega ')\). Let \({\mathcal {Q}}^o\subset {\mathbb {C}}\) denote the union of the complementary components V of the quad boundaries which are such that \({\mathbb {P}}[{\mathcal {A}}_\square (RV,R\mathcal Q)]>0\) for sufficiently large R. We assume throughout this and the next section that \(\mathcal Q^o\) has finitely many connected components and the boundaries of the quads are piecewise smooth.Footnote 12 Let \({\mathcal {I}}\) denote the sites of \({\mathbb {T}}\) which are contained in at least one quad in \(R\mathcal Q\). Throughout the section we let \(\alpha _4(R)\) be defined by \(\alpha _4(R)=\alpha ^\eta _4(8,R)\) for \(\eta =1\), where the right side is defined as in Sect. 2.3. For \(r<R\) we write \(\alpha _4(r,R)\) instead of \(\alpha ^1_4(r,R)\) since we work with lattice \(\eta =1\) throughout the appendix.

The following is the main result of this appendix. In other words, we prove that the size of the spectral sample \({\mathscr {S}}\) is of order \(R^2\alpha _4(R)\). Note that the theorem was proved in [23] for the case where \(\mathcal Q\) consists of a single quad.

Theorem B.1

$$\begin{aligned} \lim _{s\rightarrow \infty } \inf _{R>1} {\mathbb {P}}\left[ |\mathscr {S}_f| \in [s^{-1}R^2\alpha _4(R),s R^2\alpha _4(R))\cup \{0 \} \right] = 1, \end{aligned}$$

where \(|\cdot |\) denotes cardinality.

Proof

Our proof follows very closely the strategy from [23]. The main focus here is to extend the key arguments from that paper to the present multi-quads setting. In particular, the theorem will follow from Theorem B.2 and Proposition B.3 below by exactly the same argument as in the proof of [23, Theorem 7.4]. \(\square \)

Theorem B.2

Let \(U\subset \mathcal Q^o\) be open, and let \(U'\subset \overline{U'}\subset U\). Then, for some constants \(\overline{r}=\overline{r}(U',U,\mathcal Q)>0\) and \(q(U',U,\mathcal Q)>0\), for any \(r\in [\overline{r} , R\mathrm {diam}(U)]\),

$$\begin{aligned}&{{\mathbb {P}}\bigl [0<|{\mathscr {S}}_{f}\cap RU|\le r^2\,\alpha _4(r),\,{\mathscr {S}}_{f}\cap RU\subset RU'\bigr ]}\nonumber \\&\quad \le q(U',U,\mathcal Q)\, \frac{R^2\,\alpha _4(R)^2}{r^2\,\alpha _4(r)^2}\,. \end{aligned}$$
(B.1)

Proof

The theorem follows from Propositions B.5 and B.6, and from [23, Proposition 6.1]. See the proof of [23, Theorem 7.1] for a similar argument. \(\square \)

Proposition B.3

Given any \(\delta >0\) we can find an open set \(U\subset \overline{U} \subset \mathcal Q^o\), such that \({\mathbb {P}}[\mathscr {S}_{f}\subset RU]>1-\delta \).

Proof

Let \(\gamma =\bigcup _{Q\in {\mathcal {Q}}}\partial Q\subset {\mathbb {C}}\) be the union of the quad boundaries. Given \(s>0\) let \(g:\Omega \rightarrow \{-1,1 \}\) be measurable with respect to the \(\sigma \)-algebra \(\mathcal F_s\) of quad crossing information at distance \(>Rs\) from \(R\gamma \), such that

$$\begin{aligned} g(\omega ) = {\left\{ \begin{array}{ll} -1 &{}\quad \text {if } {\mathbb {P}}[f=-1\,|\,\mathcal F_s]>1/2,\\ 1 &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

By [57, Theorem 1.5], given any \(\epsilon >0\) and a quad Q it holds for all s and sufficiently small and R sufficiently large that \({\mathbb {P}}[ \epsilon<{\mathbb {P}}[ \omega (Q)\,|\,\mathcal F_s ]<1-\epsilon ]<\epsilon \), where \(\omega (Q)\in \{0,1 \}\) indicates whether Q is crossed. (Note that it is important here to assume that the boundaries of our quads are piecewise smooth.) Therefore, for sufficiently small s and sufficiently large R,

$$\begin{aligned} {\mathbb {P}}[ \epsilon<{\mathbb {P}}[ f=-1\,|\,\mathcal F_s ]<1-\epsilon ]<\epsilon . \end{aligned}$$

It follows that for sufficiently small s and sufficiently large R,

$$\begin{aligned} \begin{aligned} {\mathbb {P}}[ f\ne g ]&\le {\mathbb {P}}[ f\ne g; \epsilon< {\mathbb {P}}[f=-1\,|\,\mathcal F_s]<1-\epsilon ] \\&\quad + {\mathbb {P}}[ f=1; {\mathbb {P}}[f=-1\,|\,\mathcal F_s]>1-\epsilon ]\\&\quad + {\mathbb {P}}[ f=-1; {\mathbb {P}}[f=1\,|\,\mathcal F_s]>1-\epsilon ] < 3\epsilon , \end{aligned} \end{aligned}$$

which implies that with \(\Vert \cdot \Vert \) denoting the \(L^2\) norm we have \(\Vert f-g\Vert <10\sqrt{\epsilon }\). With \({\text {tv}}\) denoting total variation distance,

$$\begin{aligned} {\text {tv}}(\mathscr {S}_f,\mathscr {S}_g) \le \sum _{S\subset \mathcal I} |\widehat{f}(S)^2-\widehat{g}(S)^2| \le \Vert f-g\Vert \Vert f+g\Vert < 20\sqrt{\epsilon }, \end{aligned}$$

where the second inequality follows from [23, equation (2.7)]. The spectral sample of \(\mathscr {S}_g\) has distance at least Rs from \(R\gamma \), so we see that the proposition holds with \(U'\) instead of U if we let \(U'\subset {\mathbb {C}}\) be the points which have distance at least s from \(\gamma \). We have that \(\mathscr {S}\cap (U'\setminus \mathcal Q^o)=\emptyset \), and we obtain the proposition by defining \(U=U'\cap \mathcal Q^o\).

\(\square \)

Remark B.4

Note that a possibly more direct proof of this proposition would consist in decomposing the \(\alpha \)-neighborhood of the boundaries of each quads into \(O(\alpha ^{-1})\) squares of side length \(\alpha \) and then argue through a first moment bound by noticing that

$$\begin{aligned} {{\mathbb {P}}\bigl [\mathscr {S}_{f} \text { intersects the }\alpha \text {-neighbourhood of the boundaries of quads}\bigr ]} \end{aligned}$$

is dominated by the sum over all these squares of the probability that the spectral sample intersects the fixed given square. In the bulk this probability is \(O(\alpha ^{5/4+o(1)})\) and one can conclude the proof along those lines after dealing with boundary issues. In some sense such boundary issues are already dealt with in the work [57], which explains why we have chosen this other approach.

As in [23], the proof of Theorem B.2 relies on two key properties: few squares intersect the spectral sample and partial independence in the spectral sample. In the single quad case these properties are established in [23, Section 4] and [23, Section 5], respectively. The proof given in [23, Section 4] generalizes without difficulty to our multiple quads setting, while the argument in [23, Section 5] requires slightly more work. Therefore we will simply state our variant of [23, Proposition 4.2] right below, while we provide a more detailed adaption of [23, Section 5] in Sect. B.1. For a set \(S\subset {\mathbb {T}}\) and \(r>0\) define \(S_r\) to be the collection of squares in \(r{\mathbb {Z}}^2\) that intersect S.

Proposition B.5

Consider a collection \(\mathcal Q\) of finitely many quads, and let \({\mathscr {S}}\) be the spectral sample of \(f_{\mathcal Q}\). Let \(U'\subset U\subset \mathcal Q^o\), let \(\widehat{R}\) denote the diameter of U, let \(a\in (0,1)\), and suppose that the distance from \(U'\) to the complement of U is at least \(a\,\widehat{R}\). Let \({\mathcal {S}}(r,k)\) be the collection of all sets \(S\subseteq \mathcal I\) such that \(\bigl |(S\cap U)_r\bigr |=k\) and \(S\cap (U{\setminus } U')=\emptyset \). Then for \(g(k):=2^{\vartheta \log _2^2 (k+2)}\), with \(\vartheta >0\) large enough, and \(\gamma _r(\widehat{R}):=(\widehat{R}/r)^2 \alpha _4(r,\widehat{R})^2\), we have

$$\begin{aligned} \forall k,r\in {\mathbb {N}}_+\qquad {{\mathbb {P}}\bigl [{\mathscr {S}}\in {\mathcal {S}}(r,k)\bigr ]} \le c_a\,g(k)\,\gamma _r(\widehat{R})\,, \end{aligned}$$

where \(c_a\) is a constant that depends only on a and \(\mathcal Q\).

Proof

The proposition is proved by adapting the techniques of [23, Section 4]. In particular, we construct so-called annulus structures for the collection of quads \({\mathcal {Q}}\) by defining annulus structures for each component of \({\mathcal {Q}}^o\) with diameter at least \(a\widehat{R}\). \(\square \)

We also point out here that another generalization of the techniques needed here have been analyzed in the work [29], where the needed extension of [23, Section 4] happened to be more substantial and was thus written with more details.

1.1 Partial independence in the spectral sample

Proposition B.6

Let \(\mathcal Q\) be a collection of finitely many quads, and let U be an open set whose closure is contained in \(\mathcal Q^o\). For \(R>0\), let \({\mathscr {S}}:= {\mathscr {S}}_{f_{R\mathcal Q}}\) be the spectral sample of \(f_{R\mathcal Q}\), the \(\pm 1\) indicator function for the crossing event in \(R\mathcal Q\). Then, there is a constant \(\overline{r}=\overline{r}(U,\mathcal Q)\) such that for any box \(B\subset R\,U\) of radius \(r \in [\overline{r},R\mathrm {diam}(U)]\) and any set W with \(W\cap B=\emptyset \), we have

$$\begin{aligned} {{\mathbb {P}}\bigl [{\mathscr {S}}_{f_{R\mathcal Q}} \cap B' \cap {\mathcal {Z}}\ne \emptyset \bigm | {\mathscr {S}}_{f_{R\mathcal Q}}\cap B\ne \emptyset ,\, {\mathscr {S}}_{f_{R\mathcal Q}}\cap W =\emptyset \bigr ]}\ge a(U,\mathcal Q)\,, \end{aligned}$$

where \(B'\) is concentric with B and has radius r/3, the random set \({\mathcal {Z}}\) contains each element of \(\mathcal I\) independently with probability \(1/(\alpha _4(r) r^2)\), and \(a(U,\mathcal Q)>0\) is a constant that depends only on U and \(\mathcal Q\).

Proof

The proof is identical to the proof of [23, Proposition 5.11]. Propositions B.10 and B.11 below give the required first and second moment estimates. \(\square \)

Remark B.7

As we outline below, it is not too difficult to extend the proof of [23] to our present multiple-quad setting. Yet, one crucial property of our multiple-quad Boolean function \(f=f_{R\mathcal Q}\) is that it is a monotone Boolean function. Otherwise the techniques from [23] break down completely. See Remark 5.5 in [23]. This is the reason why we only control via Fourier analysis the intersection of several monotone events (crossing events) and deal with the more general ones via an inclusion-exclusion argument.

Let \(B,W\subset \mathcal I\) be disjoint. Let \(\Lambda _B=\Lambda _{f,B}\) be the event that B is pivotal for f. More precisely, \(\Lambda _B\) is the set of \(\omega \in \Omega \) such that there is some \(\omega '\in \Omega \) that agrees with \(\omega \) on \(B^c\) while \(f(\omega )\ne f(\omega ')\). Also define \(\lambda _{B,W}=\Lambda (B,W):={{\mathbb {P}}\bigl [\Lambda _B\bigm | {\mathcal {F}}_{W^c}\bigr ]}\).

The following lemma is derived as in [23, Section 5.3, equation (5.10)].

Lemma B.8

Consider the setting of Proposition B.6. Let \(\widehat{f}\) be a monotone function of a percolation configuration on \({\mathcal {I}}\) such that \(\widehat{f}\) takes values in \(\{ -1,1 \}\), and let \(\widehat{{\mathscr {S}}}\) denote the associated spectral sample. Let \(\omega ,\omega '\) be instances of critical percolation on \({\mathbb {T}}\) which are the same on \(W^c\) and independent on W. Then the following two inequalities are equivalent for any constant \(c_1>0\)

$$\begin{aligned}&{{\mathbb {P}}\bigl [ \omega ',\omega ''\in A_\square (x,\mathcal Q)\bigr ]} \ge c_1\, {{\mathbb {P}}\bigl [\omega ',\omega ''\in A_4(x,B)\bigr ]} \, {{\mathbb {P}}\bigl [ \omega ',\omega ''\in A_\square (B,\mathcal Q)\bigr ]},\\&{{\mathbb {P}}\bigl [x\in \widehat{\mathscr {S}},\ \widehat{\mathscr {S}}\cap W=\emptyset \bigr ]} \ge c_1\, {{\mathbb {E}}\bigl [\lambda _{B,W}^2\bigr ]}\, \alpha _4(r). \end{aligned}$$

Recall Definition 2.3. We call the boundary arcs \(\partial _1 Q\) and \(\partial _3 Q\) (resp. \(\partial _2 Q\) and \(\partial _4 Q\)) the open boundary arcs (resp. closed boundary arcs) of Q. For an instance of site percolation on \({\mathbb {T}}\) the quad Q is crossed (resp. not crossed) if and only if there is a path of open (resp. closed) sites connecting the two open (resp. closed) boundary arcs. See Fig. 2 for an illustration of the following lemma.

Fig. 2
figure 2

Illustration of the quads defined in Lemma B.9. The bold black boundary arcs of the quads \(Q_1,Q_2,Q_3,q_1,\ldots ,q_4,\widehat{q}_1,\widehat{q}_3\) indicate the boundary arcs which are connected on the event that the quads have an open crossing. For the quads \(Q_1\) and \(Q_2\) we are in case (b) of Lemma B.9(ii), while for the quad \(Q_3\) we are in case (a). Therefore \(A(Q_1)\) and \(Q(Q_2)\) consist of two quads each, while \(A(Q_3)\) consists of a single quad

Lemma B.9

Let V be a connected component of \(\mathcal Q^o\). For any \(V'\subset V\) such that \(\overline{V'}\subset V\), we can find quads \(q_1,\dots ,q_4,\widehat{q}_1,\widehat{q}_3\) and a collection of quads A(Q) for each \(Q\in \mathcal Q\) such the following hold.

  1. (i)

    The quads \(q_1,\dots ,q_4\) are bounded away from \(V'\) and each other, and they are contained in V. One of the open boundary arcs of \(q_1\) (resp. \(q_3\)) is equal to a boundary arc of V, while the other boundary arcs of \(q_1\) (resp. \(q_3\)) are in the interior of V. The same property holds for \(q_2\) and \(q_4\), but with closed instead of open. The quads \(q_1,\dots ,q_4\) are in counterclockwise order around \(\partial V\).

  2. (ii)

    For each quad \(Q\in \mathcal Q\) one of the following properties (a) or (b) holds.

    1. (a)

      A(Q) consists of a single quad q contained in Q, which is such that the open boundary arcs of q are contained in each of the open boundary arcs of Q.

    2. (b)

      A(Q) consists of two quads \(q',q''\) contained in Q, which are such that one open boundary arc of \(q'\) is contained in an open boundary arc of Q and the other open boundary arc of \(q'\) is contained in the closed boundary arc of either \(q_2\) or \(q_4\) which does not intersect \(\partial V\). The same property holds for \(q''\), except that \(q''\) intersects the other open boundary arc of Q.

  3. (iii)

    There is a quad \(\widehat{Q}\in {\mathcal {Q}}\) such that \(\widehat{q}_1\) and \(\widehat{q}_3\) are contained in \(\widehat{Q}\). Furthermore, one closed boundary arc of \(\widehat{q}_1\) is contained in a closed boundary arc of \(\widehat{Q}\) and the other closed boundary arc of \(\widehat{q}_1\) is contained in the open boundary arc of \(q_1\) which does not intersect \(\partial V\). The same property holds for \(\widehat{q}_3\), except that the closed boundary arcs intersect the other closed boundary arc of \(\widehat{Q}\) and an open boundary arc of \(q_4\), respectively.

  4. (iv)

    \(q_1\cup q_3\cup \widehat{q}_1\cup \widehat{q}_3\) and \(q_2\cup q_4\cup (\cup _{Q\in \mathcal Q}A(Q))\) are disjoint.

Observe that V is pivotal for \(\mathcal Q\) if all the quads in \(\{q_2,q_4\}\cup \big (\bigcup _{Q\in {\mathcal {Q}}} A(Q)\big )\) have open crossings, and none of the quads in \(\{q_1,q_3,\widehat{q}_1,\widehat{q}_3\}\) have open crossings. (N.B. obviously this is not an iff).

Proof

Choose R large and consider a percolation configuration such that there is a pivotal point \(x\in V\) for the event that all the quads in \(R\mathcal Q\) are crossed. Let x be closed. We may assume \(R^{-1}x\) is bounded away from \(\partial V\) and choose \(V'\) such that \(x\in V'\) and \(\overline{V'}\subset V\). For quads for which there is an open crossing define A(Q) and q as in (ii)(a) by using the open crossing, e.g. consider a path of open hexagons in the dual lattice connecting the two open sides of Q and let q be contained in these hexagons.

For the remaining quads \(Q\in \mathcal Q\) the vertex x is pivotal. Define \(\widehat{Q}\) to be one of these remaining quads (in Fig. 3 we chose \(\widehat{Q}=Q_2\)). If Q is a quad which is not crossed (including, among others, the particular quad \(\widehat{Q}\)), define A(Q) as in (ii)(b) by using two open arms from V to the open boundary arcs of Q. At this point we have not yet defined \(q_2\) and \(q_4\), so instead of the requirement involving \(q_2,q_4\) in (ii)(b) we assume that one of the open sides of each quad in A(Q) is contained in V. We may assume that the quads in A(Q) do not enter and exit V multiple times in the sense that for each \(q'\in A(Q)\) the set \(q'\setminus V\) has one connected component (viewing \(q'\) as a subset of \({\mathbb {C}}\)). If \(q'\) does not satisfy this property then it will hold for some quad \(\widetilde{q}\) contained in \(q'\) (such that \(\widetilde{q}\) still satisfies the requirements as specified in (ii)(b)), and we replace \(q'\) by \(\widetilde{q}\).

Define \(\widehat{q}_1,\widehat{q}_3\) satisfying (iii) by using two closed arms from V to the closed boundary arcs of \(\widehat{Q}\). Again we assume that \(\widehat{q}_1\setminus V\) and \(\widehat{q}_2\setminus V\) have one connected component.

Note that (iv) is satisfied if we let the quads in A(Q) for all \(Q\in \mathcal Q\) along with \(\widehat{q}_1,\widehat{q}_3\) be contained in the interior of the hexagons which define the crossings. Finally, we can find quads \(q_1,\dots ,q_4\) satisfying (i), (ii)(b), and (iii) (after doing local deformations of the parts of the quads in \(A(Q)\cup \{\widehat{q}_1,\widehat{q}_3 \}\) intersecting V) since \(\partial V\) can be divided into four arcs such that with these arcs in counterclockwise order, the first (resp. third) arc contains \(\widehat{q}_1\cap \partial V\) (resp. \(\widehat{q}_3\cap \partial V\)), and the union of the remaining two arcs contain \(q\cap \partial V\) for each q in some set A(Q). \(\square \)

The following is our first moment estimate. It is an analogue of [23, Proposition 5.2] for the case of multiple quads.

Proposition B.10

Consider the setup of Proposition B.6. There is a constant \(c_1>0\) (depending on U and \({\mathcal {Q}}\)) such that for any \(x\in B'\cap \mathcal I\),

$$\begin{aligned} {{\mathbb {P}}\bigl [x\in {\mathscr {S}},\ {\mathscr {S}}\cap W=\emptyset \bigr ]} \ge c_1\, {{\mathbb {E}}\bigl [\lambda _{B,W}^2\bigr ]}\, \alpha _4(r)\,. \end{aligned}$$
(B.2)

Proof

In the proof below, we will rely on the notations introduced in [23, Section 5]. It is sufficient to prove the first inequality of Lemma B.8 (which is a quasi-multiplicativity type of estimate). We will use for this the construction provided by Lemma B.9, where V is the component of \(\mathcal Q^o\) containing \(R^{-1}x\). Let \(V''\) be the connected component of U which contains the point \(R^{-1}x\), and set \(d=R{\text {dist}}(V'',V^c)\). Then define \(V'=\{y\in V\,:\, R{\text {dist}}(y,V^c)>d/3 \}\), so that \(V''\subset V'\subset V\).

Let \(\widehat{B}\subset RV\) (resp. \(\widehat{B}'\subset RV\)) be the square concentric with B of side length \(d/3+r\) (resp. \(d/6+r\)). Note that \(\widehat{B}\) has distance at least d/3 from \((RV')^c\). Let \(L_0,\dots ,L_7\) be defined as in [23, Section 5] with the annulus \(\widehat{B}\setminus \widehat{B}'\). Let E be the event \(\omega ',\omega ''\in \mathscr {A}_4(x,\widehat{B})\), with the additional requirement that the two open (resp. closed) arms cross the annulus \(\widehat{B}\setminus \widehat{B}'\) inside \(L_0,L_4\) (resp. \(L_2,L_6\)), and that there are open (resp. closed) paths that separate \(\partial \widehat{B}\cap L_j\) from \(\partial \widehat{B}'\cap L_j\) inside \(L_j\) for \(j=0,4\) (resp. \(j=2,6\)). Let \(E'\) be the event that the quads \(\bigcup _{Q\in {\mathcal {Q}}}A(Q),q_2,q_4\) rescaled by R have open crossings, and that \(q_1,q_3,\widehat{q}_1,\widehat{q}_3\) rescaled by R have closed crossings. Let \(E''\) be the event that there is an open crossing from \(\partial \widehat{B}'\) to \(R(\partial q_2\cap \partial V)\) inside \(L_0\cup ((RV)\setminus \widehat{B})\), that there is a similar crossing with \(L_4\) and \(q_4\), and that there are similar closed crossings. We have

$$\begin{aligned}&{\mathbb {P}}[ \omega ',\omega ''\in \mathscr {A}_4(x,B) ] {\mathbb {P}}[ \omega ',\omega ''\in \mathscr {A}_\square (B,R{\mathcal {Q}}) ]\nonumber \\&\quad \le {\mathbb {P}}[ \omega ',\omega ''\in \mathscr {A}_4(x,B) ] {\mathbb {P}}[ \omega ',\omega ''\in \mathscr {A}_4(B,\widehat{B}) ] \end{aligned}$$
(a)
$$\begin{aligned}&\quad \preceq {{\mathbb {P}}}[ \omega ',\omega ''\in \mathscr {A}_4(x,\widehat{B}) ]\end{aligned}$$
(b)
$$\begin{aligned}&\quad \preceq {{\mathbb {P}}}[ \omega ',\omega ''\in \mathscr {A}_4(x,\widehat{B})\cap E\cap E'\cap E'' ]\end{aligned}$$
(c)
$$\begin{aligned}&\quad \le {{\mathbb {P}}}[ \omega ',\omega ''\in \mathscr {A}_\square (x,R{\mathcal {Q}}) ]. \end{aligned}$$
(d)

Here (a) and (d) are immediate by inclusion of events, (b) is [23, Proposition 5.6], and (c) follows by using the Russo-Seymour-Welsh theorem, the FKG inequality, and compactness. \(\square \)

Fig. 3
figure 3

Illustration of the events E and \(E''\) in the proof of Proposition B.10. Open arms are blue and closed arms are orange

The following is the (easier) second moment estimate. It is an analogue of [23, Proposition 5.3] for the case of multiple quads.

Proposition B.11

Let \({\mathscr {S}}\) be the spectral sample of \(f=f_{R\mathcal Q}\), where \(\mathcal Q\) is a collection of finitely many quads. Let z be a point in one of the quads and let \(r>0\). Set \(B:= B(z,r)\) and \(B':=B(z,r/3)\). Suppose that \(B(z,r/2)\subset R\mathcal Q^o\) and that B and W are disjoint. Then for every \(x,y\in B'\cap \mathcal I\) we have

$$\begin{aligned} {{\mathbb {P}}\bigl [x,y\in {\mathscr {S}},\ {\mathscr {S}}\cap W=\emptyset \bigr ]} \le c_2\, {{\mathbb {E}}\bigl [\lambda _{B,W}^2\bigr ]}\, \alpha _4(|x-y|)\,\alpha _4(r)\,, \end{aligned}$$

where \(c_2<\infty \) is an absolute constant.

Proof

The proof is identical to the proof in [23]. Note that [23, Lemmas 2.1 and 2.2], which are used in the proof, hold for the spectral sample of general real-valued functions f of the percolation configuration. For an arbitrary set \(A\subset \mathcal I\) we let \(\Lambda _A\) be the event that A is pivotal for our quad crossing event. One key geometric argument in the proof which still holds in our setting is that if we condition on \(\omega \) restricted to the complement of \(W\cup \{x,y \}\) and if flipping \(\omega _x\) affects \(f(\omega )\), then we must have a four arm event from x to distance \(|x-y|/4\), and four arms in an annulus with outer boundary \(\partial B\) and inner boundary defined by a box centered at \((x+y)/2\) with radius \(2|x-y|\). \(\square \)

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Garban, C., Holden, N., Sepúlveda, A. et al. Liouville dynamical percolation. Probab. Theory Relat. Fields 180, 621–678 (2021). https://doi.org/10.1007/s00440-021-01057-1

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Mathematics Subject Classification

  • 82B27
  • 82B43
  • 60D05
  • 60J25
  • 60J67