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A distance exponent for Liouville quantum gravity

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Abstract

Let \(\gamma \in (0,2)\) and let h be the random distribution on \(\mathbb C\) which describes a \(\gamma \)-Liouville quantum gravity (LQG) cone. Also let \(\kappa = 16/\gamma ^2 >4\) and let \(\eta \) be a whole-plane space-filling SLE\(_\kappa \) curve sampled independent from h and parametrized by \(\gamma \)-quantum mass with respect to h. We study a family \(\{\mathcal G^\epsilon \}_{\epsilon >0}\) of planar maps associated with \((h, \eta )\) called the LQG structure graphs (a.k.a. mated-CRT maps) which we conjecture converge in probability in the scaling limit with respect to the Gromov–Hausdorff topology to a random metric space associated with \(\gamma \)-LQG. In particular, \(\mathcal G^\epsilon \) is the graph whose vertex set is \(\epsilon \mathbb Z\), with two such vertices \(x_1,x_2\in \epsilon \mathbb Z\) connected by an edge if and only if the corresponding curve segments \(\eta ([x_1-\epsilon , x_1])\) and \(\eta ([x_2-\epsilon ,x_2])\) share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014), the graph \(\mathcal G^\epsilon \) can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius n in \(\mathcal G^\epsilon \) which are consistent with the prediction of Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent \(\chi > 0\) for which the expected graph distance between generic points in the subgraph of \(\mathcal G^\epsilon \) corresponding to the segment \(\eta ([0,1])\) is of order \(\epsilon ^{-\chi + o_\epsilon (1)}\), and this distance is extremely unlikely to be larger than \(\epsilon ^{-\chi + o_\epsilon (1)}\).

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Notes

  1. Our \(\kappa \) corresponds to the parameter \(\kappa '\) in [27, 63].

  2. It is easy to see that \(\mathcal G^\epsilon \) is a planar map with the embedding given by mapping each vertex to the corresponding cell. In fact, \(\mathcal G^\epsilon \) is a triangulation provided we draw two edges instead of one between the cells \(\eta ([x_1-\epsilon , x_1])\) and \(\eta ([x_2-\epsilon ,x_2])\) whenever \(|x_1 - x_2| > \epsilon \) and these cells intersect along a non-trivial arc of each of their left and right boundaries (equivalently, both conditions in (1.3) hold); see, the introduction of [41] for more details. In this paper we only care about graph distances in \(\mathcal G^\epsilon \), so these facts will not be relevant for us.

  3. The relation between \(\widetilde{\gamma }\) and \(\gamma \) can be explained by observing that for a surface of dimension \(d_\gamma \), rescaling areas by a constant c should correspond to rescaling lengths by \(c^{1/d_\gamma }\), and we can obtain such a rescaling by replacing h by \(h+\gamma ^{-1}\log c\) and \(h+(d_\gamma \widetilde{\gamma })^{-1}\log c\), respectively.

  4. This is not the first work to apply KPZ-type results to a set which is not independent from h; the paper [7] proves a KPZ-type relation for flow lines of a GFF (in the sense of [57, 60, 61, 63]), in which case the lack of independence is much more serious and (unlike in our setting) the KPZ relation differs from the ordinary KPZ relation for independent sets.

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Acknowledgements

We thank Jian Ding, Subhajit Goswami, Jason Miller, and Scott Sheffield for helpful discussions. E.G. was supported by the U.S. Department of Defense via an NDSEG fellowship. N.H. was supported by a doctoral research fellowship from the Norwegian Research Council. X.S. was supported by the Simons Foundation as a Junior Fellow at Simons Society of Fellows. We thank two anonymous referees for helpful comments on an earlier version of this article.

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Correspondence to Ewain Gwynne.

A Proofs of some technical results

A Proofs of some technical results

Here we collect the proofs of some technical results which are used in the main body of the paper, but whose proofs are somewhat different in flavor than the main argument.

1.1 A.1 Basic estimates for the \(\gamma \)-LQG measure of a quantum cone

Here we record some basic estimates for the \(\gamma \)-LQG measure associated with an \(\alpha \)-quantum cone which are used in Sect. 3. In practice, we will always take \(\alpha =\gamma \) but it is no more difficult to treat the case of general \(\alpha \in (0,Q]\). We first have a basic lower bound for the \(\gamma \)-LQG mass of a small ball centered at 0.

Lemma A.1

Let h be a whole-plane GFF, normalized so that its circle average over \(\partial \mathbb D\) is 0 or let \(\alpha \in (0,Q]\) and let h be a circle average embedding of a \(\alpha \)-quantum cone in \((\mathbb C , 0, \infty )\) (recall Sect. 2.1.1). For \(r \in (0,1)\) and \(p > 0\),

$$\begin{aligned} \mathbb P\left[ \mu _h(B_\epsilon (z)) < \epsilon ^{2+\gamma ^2/2 +p} \right] \le \epsilon ^{ \frac{p^2}{2\gamma ^2} + o_\epsilon (1) } ,\quad \forall z\in B_r(0) ,\quad \forall \epsilon \in (0,1) \end{aligned}$$
(A.1)

with the rate of convergence of the \(o_\epsilon (1)\) depending only on p and r.

Proof

If h is a whole-plane GFF on \(\mathbb C\), normalized so that its circle average over \(\partial \mathbb D\) is 0, then the restriction of \(h-\gamma \log |\cdot |\) to \(B_1(0)\) agrees in law with the restriction to \(\mathbb D\) of the circle average embedding of a \(\gamma \)-quantum cone (see, e.g., the discussion just after [27, Definition 4.9]). Adding the function \(-\gamma \log |\cdot |\) can only increase the \(\gamma \)-LQG measure of subsets of \(\mathbb D\), so it suffices to prove (A.1) in the case when h is a whole-plane GFF.

Let \(h_\epsilon (\cdot )\) be the circle average process for h. By [34, Lemma 3.12] (c.f. [29, Lemma 4.5]), for each \(u \in (0,p)\) we have

$$\begin{aligned} \mathbb P\left[ \mu _h(B_\epsilon (z)) \le \epsilon ^{2+\gamma ^2/2+ u} e^{ \gamma h_\epsilon (z)} \right] = o_\epsilon ^\infty (\epsilon ) . \end{aligned}$$

Furthermore, \(h_\epsilon (z)\) is a centered Gaussian random variable with variance at most \(\log \epsilon ^{-1} + O_\epsilon (1)\) [29, Section 3.1], so the Gaussian tail bound implies

$$\begin{aligned} \mathbb P\left[ h_\epsilon (z) \le \frac{p - u}{\gamma } \log \epsilon \right] \le \epsilon ^{\frac{(p-u)^2}{2\gamma ^2} } . \end{aligned}$$

The statement of the lemma follows upon sending \(u\rightarrow 0\). \(\square \)

If we fix the radius of the ball, we obtain a stronger lower bound for the LQG measure.

Lemma A.2

Suppose we are in the setting of Lemma A.1. For each fixed \(r \in (0,1]\) and each \(\epsilon \in (0,1)\),

$$\begin{aligned} \mathbb P\left[ \mu _h(B_r(0) ) < \epsilon \right] = o_\epsilon ^\infty (\epsilon ) \end{aligned}$$
(A.2)

at a rate depending on r.

Proof

As in the proof of Lemma A.1, it suffices to prove the statement in the case of the whole-plane GFF. It is easy to see from [29, Lemma 4.5] (see, e.g., the proof of [34, Lemma 3.12]) that in this case \(\mathbb P[ \mu _{h}(B_r(0)) < \epsilon ^{1/2} e^{\gamma h^G_r(0)} ] = o_\epsilon ^\infty (\epsilon )\), where \(h_r(0)\) is the circle average of h over \(\partial B_r(0)\). Since \(h_r(0)\) is Gaussian with variance \(\log r^{-1}\), we also have \(\mathbb P[e^{\gamma h_r(0)} < \epsilon ^{1/2} ] = o_\epsilon ^\infty (\epsilon )\). \(\square \)

To complement the above lemmas, we also have an upper for the \(\gamma \)-LQG mass of a ball centered at 0. The proof in this case is more difficult since the logarithmic singularity at the origin increases the \(\gamma \)-LQG measure.

Lemma A.3

Let \(\alpha < Q\) (with Q as in (2.1)) and let h be a circle average embedding of an \(\alpha \)-quantum cone in \((\mathbb C , 0 , \infty )\). For \(0< p < \min \{\frac{4}{\gamma ^2} ,\frac{2}{\gamma }(Q-\alpha )\} \) and \(\epsilon \in (0,1]\),

$$\begin{aligned} \mathbb E\left[ \mu _h\left( B_\epsilon (0) \right) ^p \right] \preceq \epsilon ^{ p \left( 2 + \frac{\gamma ^2}{2} - \alpha \gamma \right) - \frac{\gamma ^2 p^2}{2} } \end{aligned}$$
(A.3)

with the implicit constant depending only on \(\alpha \) and \(\gamma \).

A similar, but stronger, estimate than Lemma A.3 is proven for the quantum sphere in [18, Lemma 3.10] (the measure studied in [18] is proven to be equivalent to the \(\gamma \)-LQG measure associated with quantum sphere in [8]). Rather than trying to deduce Lemma A.3 from this estimate, we give a direct proof.

Proof of Lemma A.3

Let \(\mathring{h} := h + \alpha \log |\cdot |\), so that by our choice of embedding \(\mathring{h}|_{\mathbb D}\) agrees in law with the restriction to \(\mathbb D\) of a whole-plane GFF. For \(r > 0\), let \(\mathring{h}_r(0)\) be the circle average of \(\mathring{h}\) over \(\partial B_r(0)\). Also let \( \mathring{h}^r := \mathring{h}(r \cdot ) -\mathring{h}_r(0)\). Then \(\mathring{h}^r|_{\mathbb D} \overset{d}{=}\mathring{h}|_{\mathbb D}\) and \(\mathring{h}^r|_{\mathbb D}\) is independent from \(\mathring{h}_r(0)\).

For \(k\in \mathbb N_0\) let \(A_k \) be the annulus \(B_{e^{-k }}(0) \setminus B_{e^{-k-1}}(0)\). By [29, Proposition 2.1],

$$\begin{aligned} \mu _h(A_k)&= \exp \left\{ - k \left( 2 + \frac{\gamma ^2}{2} - \alpha \gamma \right) + \gamma \mathring{h}_{e^{-k}} (0) \right\} \int _{A_0} |z|^{-\alpha \gamma } \, d\mu _{\mathring{h}^{ e^{-k} }}(z) . \end{aligned}$$
(A.4)

The random variable \(\mathring{h}_{e^{-k}} (0)\) is Gaussian with variance k [29, Section 3.1], so for \(p > 0\) we have

$$\begin{aligned} \mathbb E\left[ \exp \left( \gamma p \mathring{h}_{e^{-k}} (0) \right) \right] = e^{\gamma ^2 p^2 k/2 } . \end{aligned}$$
(A.5)

By [67, Theorem 2.11] and since \(\mathring{h}^{ e^{-k}}|_{\mathbb D} \overset{d}{=}\mathring{h}|_{\mathbb D}\), for each \(p \in (0, 4/\gamma ^2]\),

$$\begin{aligned} \mathbb E\left[ \left( \int _{A_0} |z|^{-\alpha \gamma } \, d\mu _{\mathring{h}^{ e^{-k} }}(z) \right) ^p \right] \preceq \mathbb E\left[ \mu _{\mathring{h}^{ e^{-k}}}(\mathbb D)^p \right] \preceq 1 . \end{aligned}$$
(A.6)

For \(0< p < \min \{1 ,\frac{2}{\gamma }(Q-\alpha )\} \), the function \(x\mapsto x^p\) is concave, hence subadditive, so summing (A.4) over all \(k \ge \lfloor \log \epsilon ^{-1} \rfloor \) and applying (A.5) and (A.6) (and recalling the independent of \(\mathring{h}^r|_{\mathbb D}\) and \(\mathring{h}_r(0)\)) gives

$$\begin{aligned} \mathbb E\left[ \mu _h\left( B_\epsilon (0) \right) ^p \right]&\le \sum _{k=\lfloor \log \epsilon ^{-1} \rfloor }^\infty \mathbb E\left[ \mu _h(A_k)^p \right] \\&\preceq \sum _{k=\lfloor \log \epsilon ^{-1} \rfloor }^{\infty } \exp \left\{ - k \left( p \left( 2 + \frac{\gamma ^2}{2} - \alpha \gamma \right) - \frac{\gamma ^2 p^2}{2} \right) \right\} \\&\preceq \epsilon ^{ p \left( 2 + \frac{\gamma ^2}{2} - \alpha \gamma \right) - \frac{\gamma ^2 p^2}{2} } . \end{aligned}$$

In the case when \(1 \le p < \min \left\{ \frac{4}{\gamma ^2} , \frac{2}{\gamma }(Q-\alpha ) \right\} \), (A.3) follows from a similar calculation with the triangle inequality for the \(L^p\) norm used in place of sub-additivity.

\(\square \)

Finally, we record an estimate for the amount of time a space-filling SLE curve parametrized by \(\gamma \)-LQG mass takes to fill in the unit disk.

Lemma A.4

Let \(\alpha < Q\) and h be a circle average embedding of an \(\alpha \)-quantum cone. Let \(\eta \) be an independent whole-plane space-filling SLE\(_{\kappa '}\) from \(\infty \) to \(\infty \) sampled independently from h, then parametrized by \(\gamma \)-quantum mass with respect to h. There exists \(c=c(\alpha ,\gamma ) > 0\) such that

$$\begin{aligned} \mathbb P\left[ \mathbb D\subset \eta ([-M,M]) \right] \ge 1 - O_M(M^{-c})\quad \text {for } M > 1. \end{aligned}$$
(A.7)

Proof

By [44, Proposition 6.2], there exists \(c_0 = c_0(\gamma ) > 0\) such that the following is true. If we let \(T_- \) (resp. \(T_+\)) be the time at which \(\eta \) starts (resp. finishes) filling in \(\mathbb D\), then for \(R >1\),

$$\begin{aligned} \mathbb P\left[ {\text {Area}}\left( \eta ([T_- , T_+]) \right) \le R \right] \ge 1 - O_R(R^{-c_0}) . \end{aligned}$$

By this and [34, Lemma 3.6], we infer that there exists \(c_1 >c_0\) such that

$$\begin{aligned} \mathbb P\left[ {\text {Area}}\left( \eta ([T_- , T_+]) \right) \subset B_R(0) \right] \ge 1 - O_R(R^{-c_1}) . \end{aligned}$$
(A.8)

By Lemma A.3 and the scaling property of the \(\gamma \)-quantum cone [27, Proposition 4.11] (see also [40, Lemma 2.2]) there exists \(b = b(\alpha ,\gamma ) >0\) and \(c_2 =c_2(\alpha ,\gamma )>0\) such that for \(M>1\),

$$\begin{aligned} \mathbb P\left[ \mu _h(B_{M^b}(0)) \le M \right] \ge 1 - O_M(M^{-c_2}) . \end{aligned}$$
(A.9)

We conclude (A.7) with \(c = c_2 \wedge (b c_1)\) by combining (A.8) (with \(R = M^b\)) and (A.9). \(\square \)

1.2 A.2 Proof of Proposition 3.5

In this subsection we will prove the KPZ relation Proposition 3.5. In fact, we will prove the following slightly more general statement, whose proof is no more difficult.

Proposition A.5

The statement of Proposition 3.5 is true with h replaced by a whole-plane GFF normalized so that its circle average over \(\partial \mathbb D\) is 0 or a circle average embedding of an \(\alpha \)-quantum cone for \(\alpha < Q\). In the case of the whole-plane GFF, one in fact has the slightly stronger estimate

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0} \frac{\log \mathbb E[N^\epsilon ]}{\log \epsilon ^{-1}} \le \widehat{d}_\gamma . \end{aligned}$$
(A.10)

We believe that (A.10) is also true for the \(\alpha \)-quantum cone, but our proof yields only the weaker bound (3.15) in this case.

For the proof of Proposition A.5, we will use the following notation. For \(\delta >0\), let \(\mathfrak S_\delta \) be the set of closed squares with side length \(\delta \) and endpoints in \(\delta \mathbb Z^2\). For \(z\in \mathbb C\) let \(S_\delta (z)\) denote the element of \(\mathfrak S_\delta \) which contains z; \(S_\delta (z)\) is uniquely defined except if one or both of the coordinates of z is a multiple of \(\delta \), in which case we make an arbitrary choice between the \(\le 4\) possibilities when defining \(S_\delta (z)\). We note that in the terminology of Proposition 3.5,

$$\begin{aligned} N^\delta = \#\left\{ S\in \mathfrak S_\delta : S\cap X\not =\emptyset \right\} . \end{aligned}$$
(A.11)

We will deduce Proposition A.5 from a variant of the proposition corresponding to an alternative notion of quantum dimension which is closely related to the box-counting dimension considered in [29] but involves squares of side length \(\delta \) which intersect X and whose \(\delta \)-neighborhoods have quantum mass at most \(\epsilon \), rather than squares which themselves have quantum mass at most \(\epsilon \). Let \(X\subset D\subset \mathbb C\) be a random set as above. Let h be either a whole-plane GFF with additive constant chosen so that the circle average of h over \(\partial \mathbb D\) is zero, or a zero boundary GFF in a bounded domain \(\widetilde{D}\subset \mathbb C\) satisfying \(\overline{D}\subset \widetilde{D}\). For \(S\in \mathfrak S_\delta \) the \(\delta \)-neighborhood \(\widetilde{S}\) of S is defined by

$$\begin{aligned} \widetilde{S}=\{z\in \mathbb C\,:\,{\text {dist}}(z,S)<\delta \}. \end{aligned}$$
(A.12)

We define the dyadic parent\(S_-\) of S be the unique element of \(\mathfrak S_{2\delta }\) containing S. For \(\epsilon >0\) we define a \((\mu _h,\epsilon )\)-box to be a dyadic square \(S\in \cup _{k\in \mathbb Z} \mathfrak S_{2^{-k}}\) which satisfies (in the notation introduced just above) \(\mu _h(\widetilde{S})<\epsilon \) and \(\mu _h(\widetilde{S}_-)\ge \epsilon \). In the case of the zero boundary GFF we extend the measure \(\mu _h\) to a measure on \(\mathbb C\) by assigning measure 0 to the complement of \(\widetilde{D}\). Let \(\mathfrak S^\epsilon \) be the set of \((\mu _h,\epsilon )\)-boxes. Since \(\mu _h\) is non-atomic, for each \(z\in \mathbb C\) and \(\epsilon >0\) for which none of the coordinates are dyadic, there is a unique square \(S^\epsilon (z) \in \mathfrak S^\epsilon \) which contains z; in the case where one or both of the coordinates is dyadic we define \(S^\epsilon (z)\) uniquely by also requiring that \(S^\epsilon (z)=S_\delta (z)\) for some \(\delta =2^{-k}\), \(k\in \mathbb Z\), where \(S_\delta (z)\) is defined as in the beginning of this section. Note that the difference between our notion of a \((\mu _h,\epsilon )\)-box, and the notion of a \((\mu _h,\epsilon )\)-box considered in [29, Section 1.4], is that we consider dyadic squares where the neighborhood of each square has a certain quantum measure, instead of considering the measure of the squares themselves. See Fig. 9 for an illustration.

For \(\epsilon >0\) define \(\widehat{N}^\epsilon =\widehat{N}^\epsilon (X)\) to be the number of \((\mu _h,\epsilon )\)-boxes needed to cover X, i.e.,

$$\begin{aligned} \widehat{N}^\epsilon = \#\{S\in {\mathfrak S}^\epsilon \,:\, S\cap X\ne \emptyset \} . \end{aligned}$$

The box quantum expectation dimension of X, if it exists, is the limit

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\widehat{N}^\epsilon ]}{\log \epsilon ^{-1}} \in [0,1] . \end{aligned}$$
(A.13)

The following lemma is a version of [29, Proposition 1.6] with our alternative notion of a \((\mu _h,\epsilon )\)-box. Recall that we assume h is either a whole-plane GFF with unit circle average zero, or a zero boundary GFF.

Lemma A.6

If the Euclidean expectation dimension \(\widehat{d}_0\) of X exists and X is independent of h, then the box quantum expectation dimension of X exists and is given by \(\widehat{d}_\gamma \), where \(\widehat{d}_\gamma \in [0,1]\) solves (3.13).

Fig. 9
figure 9

The set of \((\mu _h,\epsilon )\)-boxes on the figure is the set of squares which do not contain any smaller squares. The figure illustrates various neighborhoods associated with \(z\in \mathbb C\). The quantum dimension of a random fractal X is defined in terms of the number of squares \(S^\epsilon (z)\) needed to cover X. The set \(\widetilde{S}^\epsilon (z)\) is a neighborhood of \(S^\epsilon (z)\), while \(\widetilde{S}^\epsilon _-(z)\) is a neighborhood of the dyadic parent \(S^\epsilon _-(z)\) (which is not labelled on the figure) of \(S^\epsilon (z)\). The square \(S^\epsilon (z)\) is defined such that \(\mu _h(\widetilde{S}^\epsilon )<\epsilon \) and \(\mu _h(\widetilde{S}^\epsilon _-)\ge \epsilon \)

Proof

First we consider the case where h is a zero boundary GFF on \(\widetilde{D}\). It is sufficient to establish the following two inequalities

$$\begin{aligned} \liminf _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\widehat{N}^\epsilon ]}{\log \epsilon ^{-1}}\ge \widehat{d}_\gamma \quad {\text {and}} \quad \limsup _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\widehat{N}^\epsilon ]}{\log \epsilon ^{-1}}\le \widehat{d}_\gamma . \end{aligned}$$
(A.14)

The first inequality of (A.14) is immediate, since the number of boxes \(\widehat{N}^\epsilon \) in our cover is at least as large as the number of boxes in the cover considered in [29, Proposition 1.6], since each \((\mu _h,\epsilon )\)-box with our definition is contained in a \((\mu _h,\epsilon )\)-box with the definition considered in [29].

We will now establish the second inequality of (A.14). Let \(\mathfrak S^\epsilon _-\) denote the set of dyadic parents of squares in \(\mathfrak S^\epsilon \). With \(\widetilde{S}\) as in (A.12), define the following quantum \(\epsilon \)-neighborhoods of X:

$$\begin{aligned} \widetilde{S}^\epsilon (X) := \bigcup _ { S\in \mathfrak S^\epsilon \,:\, S\cap X\ne \emptyset } \widetilde{S},\quad \widetilde{S}_-^{\epsilon }(X) := \bigcup _{ S\in \mathfrak S_-^\epsilon \,:\, S\cap X\ne \emptyset } \widetilde{S}. \end{aligned}$$

For \(z\in \mathbb C\) define \(S^\epsilon _-(z)\) to be the dyadic parent of \(S^\epsilon (z)\), and define \(\widetilde{S}^\epsilon (z)\) (resp. \(\widetilde{S}^\epsilon _-(z)\)) to be the \(\delta \)-neighborhood of \(S^\epsilon (z)\) (resp. the \(2\delta \)-neighborhood of \(S_-^\epsilon (z)\)), where \(\delta \) is the side length of \(S^\epsilon (z)\). The first step of our proof is to reduce the lemma (for the case of a zero boundary GFF) to proving the following estimate:

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\mu _h(\widetilde{S}_-^{\epsilon }(X))]}{\log \epsilon ^{-1}} \le \widehat{d}_\gamma -1. \end{aligned}$$
(A.15)

Let

$$\begin{aligned} \widehat{N}^\epsilon _- = \# \{S \in \mathfrak S_-^{\epsilon } \,:\, S \cap X\ne \emptyset \} . \end{aligned}$$

By (A.16), which we will explain just below, we see that (A.15) is sufficient to prove the lemma:

$$\begin{aligned} \begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\epsilon \widehat{N}^\epsilon ]}{\log \epsilon ^{-1}} = \lim _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\epsilon \widehat{N}_-^\epsilon ]}{\log \epsilon ^{-1}} \le \lim _{\epsilon \rightarrow 0} \frac{\log \mathbb E[\mu _h(\widetilde{S}_-^{\epsilon }(X))]}{\log \epsilon ^{-1}}. \end{aligned} \end{aligned}$$
(A.16)

The first equality of (A.16) follows by \(\widehat{N}^\epsilon _-\le \widehat{N}^\epsilon \le 4 \widehat{N}^\epsilon _-\). The second estimate of (A.16) follows since for any \(z\in D\) it holds that \(\mu _h(\widetilde{S}_-^{\epsilon }(z))\ge \epsilon \) and \(\mu _h(\widetilde{S}^{\epsilon }_-(z)\cap S)>0\) for at most 9 of the dyadic squares \(S\in \mathfrak S_-^{\epsilon }\) which intersect X.

Our justification of (A.15) will be very brief, since a similar argument can be found in [29]. Let \(\Theta =\mathcal Z^{-1}e^{\gamma h}\,dz\,dh\) be the rooted probability measure defined in [29, Section 3.3]. Proceeding similarly as in [29], and letting \(\delta =\delta (z,\epsilon )\) denote the (random) side length of \(S^\epsilon (z)\) for \((z,h)\sim \Theta \), we see that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{\mathbb E[\mu _h(\widetilde{S}^\epsilon _-(X))]}{\log \epsilon ^{-1}} = \lim _{\epsilon \rightarrow 0}\frac{\mathbb P [\widetilde{S}_-^\epsilon (z)\cap X\ne \emptyset ]}{\log \epsilon ^{-1}} = \lim _{\epsilon \rightarrow 0}\frac{\mathbb E [\delta ^{2-\widehat{d}_0 }]}{\log \epsilon ^{-1}} \le \widehat{d}_\gamma -1. \quad \end{aligned}$$
(A.17)

In particular, the first equality of (A.17) follows by the argument right after the statement of [29, Theorem 4.2], and the second equality of (A.17) follows by the argument of the first paragraph in the proof of [29, Theorem 4.2]. The last inequality of (A.17) follows by using that the dyadic squares \(S^\epsilon (z)\) have a side length which is smaller than or equal to the corresponding dyadic squares considered in [29, Section 1.4], and that the last inequality of (A.17) holds for the dyadic squares considered in [29]. The estimate (A.17) implies (A.15), which concludes the proof of the lemma for the case of the zero boundary GFF.

Next we consider the case where h is a whole-plane GFF with additive constant chosen so that its circle average over \(\partial \mathbb D\) is 0. Choose \(R>0\) such that \(D \subset B_{R/4}(0)\). Then \(h|_{B_R(0)}=h^0+\mathfrak h\), where \(h^0\) is a zero boundary GFF in \(B_R(0)\), and \(\mathfrak h\) is a harmonic function in \(B_R(0)\). By [34, Lemma 3.12], and by using that \(\mathfrak h(0)\) is the circle average of h around \(B_R(0)\), so that \(\mathfrak h(0)\) is a centered Gaussian random variable with variance \(\log (R)\), for any \(u >0\),

$$\begin{aligned} \mathbb P\left[ \sup _{z\in B_{R/2}(0)}|\mathfrak h(z)| > u \log \delta ^{-1}\right] = o_\delta ^\infty (\delta ). \end{aligned}$$

Hence, except on an event of probability \(o_\delta ^\infty (\delta )\), for any \(A\subset B_{R/2}(0)\), we have \(\delta ^{\gamma u}\mu _{h^0}(A)\le \mu _h(A)\le \delta ^{-\gamma u}\mu _{h^0}(A)\). Since \(u>0\) is arbitrary, the statement of the lemma for the case of a whole-plane GFF follows from the case of a zero-boundary GFF on \(\widetilde{D} = B_R(0)\). \(\square \)

We will apply the following basic lemma in our proof of Proposition A.5.

Lemma A.7

Let \(\mu _h\) be the \(\gamma \)-LQG measure associated with a whole-plane GFF with additive constant chosen such that the average around \(\partial \mathbb D\) is 0. Let \(D\subset \mathbb C\) be a bounded open set. For \(\delta >0\) let \(\mathcal B_\delta \) be a deterministic collection of at most \( \delta ^{-2}\) Euclidean balls of radius \(\delta >0\) contained in D, and define \(A_\delta :=\max _{B\in \mathcal B_\delta }\mu _h(B)\). Given any \(M>0\) we can find \(s =s(M) >0\) such that \(\mathbb P(A_\delta >\delta ^{s})\preceq \delta ^{sM}\), where the implicit constant is independent of \(\delta \).

Proof

By [34, Lemma 5.2] and the Chebyshev inequality, for any \(B\in \mathcal B_\delta \), \(\beta \in [0,4/\gamma ^2)\) and \(s>0\), we have

$$\begin{aligned} \mathbb P\left[ \mu _h(B)>\delta ^{s} \right] \le \delta ^{f(\beta )-s\beta +o_\delta (1)}, \end{aligned}$$

where \(f(\beta )=(2+\frac{\gamma ^2}{2})\beta -\frac{\gamma ^2}{2}\beta ^2\). By the union bound,

$$\begin{aligned} \mathbb P\left[ A_\delta>\delta ^{s} \right] \preceq \delta ^{-2} \max _{B\in \mathcal B_\delta } \mathbb P\left[ \mu _h(B)>\delta ^{s} \right] \le \delta ^{f(\beta )-2-s\beta +o_\delta (1)}. \end{aligned}$$

If we choose \(\beta \in (1,4/\gamma ^2)\) then for small enough \(s>0\), we have \(f(\beta )-2-s\beta >sM\), which implies the lemma. \(\square \)

Proof of Proposition A.5

First we consider the case where h is a whole-plane GFF normalized as in the statement of the proposition. Fix a large constant \(C>0\) to be chosen later, depending only on \(\gamma \). For \(\epsilon ,u>0\) let \(E^u_\epsilon \) be the event that the following is true.

  1. (i)

    All squares \(S\in \mathfrak S^\epsilon \) for which \(S\cap D\ne \emptyset \) have Euclidean side length at least \(\epsilon ^K\), where \(K>0\) is chosen sufficiently large such that the probability of this event is at least \(1-\epsilon ^{C}\). Existence of an appropriate K (independent of \(\epsilon ,u\)) follows from Lemma A.7 applied with e.g. \(\delta =10\epsilon ^K\), \(M=1000\), and \(\mathcal B_\delta \) a collection of balls such that each \(S\in \widetilde{\mathfrak S}_{2\delta }\) for which \(S\cap D\ne \emptyset \) is contained in a ball in \(\mathcal B_\delta \), where \(\widetilde{\mathfrak S}_{2\delta }\) is the set of \(2\delta \)-neighborhoods of boxes in \(\mathfrak S_{2\delta }\).

  2. (ii)

    For any interval \(I\subset \mathbb R\) for which \(\delta :={\text {diam}}(\eta (I))<\epsilon ^u\) and \(\eta (I) \cap D \not =\emptyset \) the set \(\eta (I)\subset \mathbb C\) contains a ball of radius at least \(\delta ^{1+u}\).

By [34, Proposition 3.4 and Remark 3.9], the probability of the event in (ii) is of order \(1-o_\epsilon ^\infty (\epsilon )\), at a rate depending only on u and the diameter of D. Hence \(\mathbb P\big ((E^u_\epsilon )^c\big )\preceq \epsilon ^{C}\). If the event (ii) occurs, then for any dyadic box S of side length \(\delta \in (0,\epsilon ^K)\), the number of disjoint SLE segments \(\eta (I)\) for \(I\subset \mathbb R\) any interval which intersect both S and \(\mathbb C\setminus \widetilde{S}\) is bounded by \(\delta ^{-2u}\) (c.f. [34, Lemma 5.1]). Therefore the condition (i) implies that on the event \(E^u_\epsilon \) we have \(N^\epsilon \preceq \epsilon ^{-2K u} \widehat{N}^\epsilon \). Note that \(N^\epsilon \preceq \mu _h(\widetilde{D}) \epsilon ^{-1}\) for \(\widetilde{D}\) a slightly larger open set containing \(\overline{D}\), so by Hölder’s inequality and the moment estimate in [67, Theorem 2.11], we see that \(\mathbb E[{\mathbb {1}}_{(E_\epsilon ^u)^c} N^\epsilon ]\) decays faster than any power of \(\epsilon \). It follows that

$$\begin{aligned} \mathbb E[N^\epsilon ] \preceq \mathbb E[\epsilon ^{-2Ku} \widehat{N}^\epsilon ] + \mathbb E[{\mathbb {1}}_{(E_\epsilon ^u)^c} N^\epsilon ] \preceq \epsilon ^{-2K u} \mathbb E[\widehat{N}^\epsilon ]. \end{aligned}$$

Since \(u>0\) was arbitrary, an application of Lemma A.6 concludes the proof of the proposition for the case of h a whole-plane GFF.

Now we assume h is the circle average embedding of an \(\alpha \)-quantum cone and that D lies at positive distance from 0. Let \(\widetilde{D} \) be a slightly larger domain containing \(\overline{D}\) which also lies at positive distance from 0. By [34, Lemma 3.10], we can couple h with an instance of a whole-plane GFF \(h^G\) (normalized as above) satisfying the following property. There is a constant \(c = c(\gamma , \alpha ) >0\) such that for each \(u > 0\), it holds except on an event of probability \(\preceq \epsilon ^{ cu}\) that \(\epsilon ^{u/3} \mu _h(A) \le \mu _{h^G}(A) \le \epsilon ^{-u/3} \mu _h(A)\) for each \(A\subset \widetilde{D}\). Let \(N^\epsilon \) (resp. \(N^\epsilon _G\)) denote the number of boxes in (3.14) when the field is h (resp. \(h^G\)). By the coupling between h and \(h^G\), except on an event of probability \(\epsilon ^{cu}\), we have \(N^\epsilon \le 10 N^{\epsilon ^{1+u/3}}_G\). We conclude the proof of the proposition by using the above result for the whole-plane GFF (and we decrease c in the very last step if necessary):

$$\begin{aligned} \mathbb P\left[ N^\epsilon>\epsilon ^{-u-d_\gamma }\right] \preceq \mathbb P\left[ N^\epsilon> 10 N_G^{\epsilon ^{1+u/3}} \right] + \mathbb P\left[ 10 N^{\epsilon ^{1+u/3}}_G > \epsilon ^{-u-d_\gamma }\right] \preceq \epsilon ^{cu}. \end{aligned}$$

\(\square \)

1.3 A.3 Proof of Lemma 5.3

In this section we prove our restricted sub-addivity lemma, Lemma 5.3. The following recursive relation is the key observation for the proof.

Lemma A.8

Assume we are in the setting of Lemma 5.3. For each \(n,m \in \mathbb N\) with \( n^p \le m \le \lambda n\), we have

$$\begin{aligned} a_n \le \frac{n}{m} a_m + C( \lambda ^{-1} + 1) m + C \frac{n^{1+p} }{m} . \end{aligned}$$

Proof

Let \(k_* := \lfloor n/m - \lambda ^{-1} \rfloor \) be the largest \(k\in \mathbb N\) for which \(n - km \ge \lambda ^{-1} m\). Note that \(k_* \le n/m - \lambda ^{-1} \). By the subaddivity hypothesis (5.4), for each \(k \in [0,k_*]_{\mathbb Z}\) we have

$$\begin{aligned} a_{n-(k-1)m} \le a_m + a_{n-k m} + C(n-km)^p . \end{aligned}$$

By iterating this estimate \(k_*\) times we get

$$\begin{aligned} a_n \le k_* a_m + a_{n-k_* m} + C k_* n^p . \end{aligned}$$
(A.18)

We have \(k_* \le n/m\) and by maximality of \(k_*\) we have \(n - k_* m \le (\lambda ^{-1} + 1) m\) so our sub-linearity hypothesis (5.5) implies \(a_{n-k_* m} \le C ( \lambda ^{-1} + 1) m\). Thus the statement of the lemma follows from (A.18). \(\square \)

Lemma A.9

Let \(f ,g : \mathbb N \rightarrow \mathbb N\) be non-decreasing functions and suppose there exists \(n_0\in \mathbb N\) such that \(f(n) > n\) and \(g(n) \ge f(f(n))\) for \(n\ge n_0\). Let \(\{b_n\}_{n\in \mathbb N}\) be a sequence of real numbers and suppose there exists a \(\chi >0\) with the following property. For each sequence \(\{n_k\}_{k\in \mathbb N}\) with \(n_k\rightarrow \infty \) and \(f(n_k) \le n_{k+1} \le g(n_k) \) for each \(k\in \mathbb N\), we have \(\lim _{k\rightarrow \infty } b_{n_k} = \chi \). Then \(\lim _{n\rightarrow \infty } b_n = \chi \).

Proof

For \(r \in \mathbb N\), let \(f^r\) and \(g^r\) be the n-fold compositions of f and g, respectively.

Suppose that \(\{m_j \}_{j\in \mathbb N}\) is an increasing sequence of positive integers with \(m_1 \ge n_0\) and \(m_{j+1} \ge g(m_j)\) for each \(j\in \mathbb N\). We claim that \(\lim _{j\rightarrow \infty } b_{m_j} = \chi \). To see this, we will construct a sequence \(\{n_k\}_{k\in \mathbb N}\) with \(f(n_k) \le n_{k+1} \le g(n_k) \) for each \(k\in \mathbb N\) such that \(\{m_j\}_{j\in \mathbb N}\) is a subsequence of \(\{n_k\}_{k\in \mathbb N}\). Let \(r_1 = 1\) and for \(j\ge 2\), let \(r_j \) be chosen so that \(f^{r_j}(m_{j-1} ) \le m_j < f^{r_j+1}(m_{j-1})\). Such an \(r_j\) exists since \(m_{j-1} \ge n_0\) so \(f^r(m_{j-1}) \ge f^{r-1}(m_{j-1}) + 1\) for \(r \in \mathbb N\), whence \(\lim _{r\rightarrow \infty } f^r(m_{j-1}) = \infty \).

Since \(m_j \ge g(m_{j-1}) \ge f^2(m_{j-1})\) we have \(r_j \ge 2\). Therefore \(f^{r_j-1}(m_{j-1} ) \ge m_{j-1} \ge n_0\). By definition of \(r_j\) and \(g(n)\ge f(f(n))\) for \(n\ge n_0\),

$$\begin{aligned} g(f^{r_j-1}(m_{j-1}) ) \ge f^{r_j+1}(m_{j-1}) \ge m_j . \end{aligned}$$
(A.19)

For \(j\in \mathbb N\), let \(k_j := \sum _{i=1}^j r_j\). Let \(n_1 := m_1\). For \(j \ge 2\) and \(k \in (k_{j-1} , k_j)_{\mathbb Z}\), let \(n_{k} := f^{k-k_{j-1}}(m_{j-1})\). Let \(n_{k_j} := m_j\). We claim that \(f(n_k) \le n_{k+1} \le g(n_k) \) for each \(k\in \mathbb N\). Indeed, given \(k\in \mathbb N\), let \(j \in \mathbb N\) be chosen so that \(k \in [k_{j-1} , k_j-1]_{\mathbb Z}\). If \(k \not =k_j-1\), then we have \(n_{k+1} = f(n_k)\), so clearly the desired inequalities hold in this case. If \(k = k_j-1\), then we have \(n_{k+1} = m_j\) and \(n_k = f^{k_j-k_{j-1} -1}(m_{j-1}) = f^{r_j-1}(m_{j-1})\). By (A.19) we have \(g(n_k) \ge n_{k+1}\) and by definition of \(r_j\) we have \(f(n_k) \le n_{k+1}\), as required. Since \(\lim _{k\rightarrow \infty } b_{n_k} = \chi \) (by hypothesis) we also have \(\lim _{j\rightarrow \infty } b_{m_j} = \chi \).

We now argue that \(\lim _{n\rightarrow \infty } b_n = \chi \). If not, we can find an increasing sequence \(m_j \rightarrow \infty \) and \(\epsilon > 0\) such that \(|b_{m_j} - \chi | \ge \epsilon \) for each \(j\in \mathbb N\). By passing to a subsequence we can arrange that \(m_1 \ge n_0\) and \(m_{j+1} \ge g(m_j)\) for each \(j\in \mathbb N\). Then the claim above implies that \(\lim _{j\rightarrow \infty } b_{m_j} = \chi \), which is a contradiction. \(\square \)

Proof of Lemma 5.3

Fix \(q \in (1,p^{-1/4})\) and \(\widehat{q} \in (q^2 ,p^{-1/2})\). For \(n\in \mathbb N\) let \(f(n) := \lceil n^q \rceil \) and \(g(n):= \lfloor n^{\widehat{q} } \rfloor \). Observe that f and g satisfy the hypotheses of Lemma A.9. By Lemma A.9 it suffices to show that there is a \(\chi \in \mathbb R\) such that for each sequence \(\{n_k\}_{k\in \mathbb N}\) with \(n_1 \ge 2\) and \( n_k^q \le n_{k+1} \le n_k^{\widehat{q}} \) for each \(k\in \mathbb N\), we have \(\lim _{k\rightarrow \infty } a_{n_k}/n_k = \chi \).

Fix such a sequence \(\{n_k\}_{k\in \mathbb N}\) and let \(b_k := a_{n_k}/n_k\). Since \(\widehat{q} < p^{-1 }\), there is a \(k_0 \in \mathbb N\) such that for \(k\ge k_0\), we have \(n_{k+1}^{p } \le n_k \le \lambda n_{k+1}\). By Lemma A.8, for \(k\ge k_0\) we have

$$\begin{aligned} a_{n_{k+1} } \le \frac{n_{k+1} }{n_k} a_{n_k} + C ( \lambda ^{-1} + 1) n_k + C \frac{n_{k+1}^{p+1} }{n_k} \end{aligned}$$

Dividing by \(n_{k+1}\) gives

$$\begin{aligned} b_{k+1} \le b_k + u_k ,\quad {\text {where}}\; u_k := C(\lambda ^{-1} + 1) \frac{n_k}{n_{k+1}} + \frac{n_{k+1}^p}{n_k} . \end{aligned}$$
(A.20)

Since \(n_1 \ge 2\) and \(n_k^q \le n_{k+1} \le n_k^{\widehat{q}}\) for each \(k\in \mathbb N\) we have \(n_k \ge 2^{q^{k-1}}\) for each \(k\in \mathbb N\) and

$$\begin{aligned} u_k \le C ( \lambda ^{-1} + 1) n_k^{-(q-1)} + n_k^{-(1- \widehat{q} p )} \le O_k(1) \left( 2^{-(q-1) q^{k-1}} + 2^{-(1-\widehat{q} p) q^{k-1}} \right) . \end{aligned}$$

Since \(1< q< \widehat{q} < p^{-1/2}\), this is summable. Let

$$\begin{aligned} \widetilde{b}_k := b_k - \sum _{j=1}^{k -1} u_j \quad {\text {and}} \quad \beta := \sum _{j=1}^\infty u_j . \end{aligned}$$

The relation (A.20) implies that \(\widetilde{b}_{k+1} \le \widetilde{b}_k\) for each \(k\in \mathbb N\). Since \(\widetilde{b}_k \ge -\beta \) for each k, we infer that \(\lim _{k\rightarrow \infty } \widetilde{b}_k\) exists. Hence also

$$\begin{aligned} \chi := \lim _{k\rightarrow \infty } b_k = \lim _{k\rightarrow \infty } \widetilde{b}_k + \beta \end{aligned}$$

exists. It remains to show that the \(\chi \) does not depend on the initial choice of sequence \(\{n_k\}_{k\in \mathbb N}\). To this end, it is enough to show that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{ a_n }{n } \le \chi , \end{aligned}$$
(A.21)

since then the limiting values \(\chi \) arising from two different choices of subsequence must agree by symmetry.

To prove (A.21), suppose given \(n\in \mathbb N\) with \(n\ge n_{k_0+2}\) (with \(k_0\) defined as in the beginning of the proof). Let \(k \in \mathbb N\) be the largest integer such that \(n_{k+1} \le n\), and note that \(k\ge k_0\). Then our condition on the \(n_k\)’s implies that \(n^{1/\widehat{q}^2} \le n_k \le n^{1/q}\). Since \(1/q < 1\) and \(1/\widehat{q}^2 >p\), Lemma A.8 with \(m = n_k\) implies that

$$\begin{aligned} a_n&\le \frac{n}{n_k} a_{n_k} + C (\lambda ^{-1} + 1) n_k + C \frac{n^{1+p}}{n_k} \\&\le \chi n(1+o_n(1)) + C (\lambda ^{-1}+1) n^{1/q} + C n^{1 + p - 1/\widehat{q}^2} = \chi n(1+o_n(1)) . \end{aligned}$$

\(\square \)

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Gwynne, E., Holden, N. & Sun, X. A distance exponent for Liouville quantum gravity. Probab. Theory Relat. Fields 173, 931–997 (2019). https://doi.org/10.1007/s00440-018-0846-9

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