Heat Kernel for Liouville Brownian Motion and Liouville Graph Distance

Abstract

We show the existence of the scaling exponent $\chi = \chi (\gamma )$, with

$$\begin{aligned} 0 < \chi \le \frac{4}{\gamma ^2} \left( \left( 1+ {\gamma ^2} / 4 \right) - \sqrt{1+ {\gamma ^4} / {16} } \right) , \end{aligned}$$

of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $\gamma <2$ on ${\mathbb {V}} =[0,1]^2$. We also show that the Liouville heat kernel satisfies, for each fixed $u,v\in {\mathbb {V}}^o$, the short time estimate

$$\begin{aligned} \lim _{t\rightarrow 0} \frac{\log |\log {\textsf {p} }_t^\gamma (u,v)| }{|\log t|}=\frac{\chi }{2-\chi }, \ \mathrm{a.s..} \end{aligned}$$

Appendix

In this appendix, we record, for use in subsequent work, a few lemmas that can be readily deduced from the techniques employed in this paper; these lemmas are not used in the paper. Let $\lambda = \frac{1}{20}$ as in Lemma 5.4. Denote $\bar{{\mathbb {V}}}_u = {{\mathbb {V}}}_{u, \lambda }$ and $\bar{{\mathbb {V}}}_{u, \alpha } = {{\mathbb {V}}}_{u, \alpha \lambda }$ for $\alpha \in (0,1)$.

Lemma 6.1

Fix $\alpha \in (0, 1)$. Let $\chi$ be as in Lemma 5.3. Then, for any $u\in \bar{{\mathbb {V}}}$,

$$\begin{aligned}&\lim _{\delta \rightarrow 0} \frac{{{\mathbb {E}}}\log \left( \min _{x\in \partial \bar{{\mathbb {V}}}_{u, \alpha }, \ y\in \partial \bar{{\mathbb {V}}}_u}D_{\gamma , \delta }(x, y) \right) }{\log \delta ^{-1}}\nonumber \\&\quad = \lim _{\delta \rightarrow 0} \frac{{{\mathbb {E}}}\log \left( \min _{x\in \partial \bar{{\mathbb {V}}}_{u, \alpha }, \ y\in \partial \bar{{\mathbb {V}}}_u} D_{\gamma , \delta , \eta }(x, y) \right) }{\log \delta ^{-1}} = \chi . \end{aligned}$$

(137)

Proof

The first equality holds due to Lemma 3.10 and the main task is to prove the second equality. By Lemma 5.4 and a similar derivation to (124), we get that for any $\kappa >0$, $v\in {\mathbb {V}}$

$$\begin{aligned} {{\mathbb {E}}}\log \left( \min _{ y\in \partial {{\mathbb {V}}}_{v, \kappa }} D_{\gamma , \delta , \eta }(v, y) \right) = \big ( \chi +o(1) \big ) \log \delta ^{-1}. \end{aligned}$$

(138)

Thus it suffices to prove a lower bound in (137). The proof is similar to that of Lemma 5.4.

By Proposition 3.17, it suffices to show that for any fixed $\iota >0$ and any segment $L_\delta \subseteq \partial \bar{{\mathbb {V}}}_{u, \alpha }$ with length in $[\delta ^{2\iota }/2, \delta ^{2\iota }]$ we have

$$\begin{aligned} {{\mathbb {E}}}\log \left( \min _{x\in L_\delta , y\in \partial \bar{{\mathbb {V}}}_u} D_{\gamma , \delta , \eta }(x, y) \right) \ge (\chi - 2\iota ) \log \delta ^{-1}. \end{aligned}$$

Suppose the preceding statement fails for some $L_\delta$. Let $v_{ L_\delta }$ be an arbitrary point on $L_\delta$. As shown in Fig. 3 employed in the proof of Lemma 5.4, we can construct four short crossings through four rectangles (with dimension $10| L_\delta | \times 40 | L_\delta |$) which altogether form a contour enclosing $L_\delta$. Consequently, the union of these short crossings, the geodesic between $L_\delta$ and $\partial \bar{{\mathbb {V}}}_u$, as well as the geodesic between $v_{ L_\delta }$ and $\partial \bar{{\mathbb {V}}}_u$ contains a path between $v_{ L_\delta }$ and $\partial \bar{{\mathbb {V}}}_u$. Therefore, by the same argument as in Lemma 5.4, we get that

$$\begin{aligned} {{\mathbb {E}}}\log \left( \min _{y\in \partial \bar{{\mathbb {V}}}_u}D_{\gamma , \delta , \eta }(v_{ L_\delta }, y) \right) \le (\chi - \iota ) \log \delta ^{-1}. \end{aligned}$$

This contradicts with (138). Thus, we complete the proof of the lemma by contradiction.

$\square$

Fix $\xi >0$ through out the appendix. For any Euclidean ball B, we denote by 2B a Euclidean ball concentric with B, whose radius is double that of B. For $\delta >0$ and any two distinct points $u, v\in {{\mathbb {V}}}^\xi$, we define a variation of Liouville graph distance $D^{({\textsf {2} })}_{\gamma , \delta , \xi }(u, v)$ to be the minimal d such that there exist Euclidean balls $B_1, \ldots , B_d \subseteq {\mathbb {V}}^\xi$ with rational centers and $M_\gamma (2B_i)\le \delta ^2$ for $1\le i\le d$, whose union contains a path from u to v.

For an Euclidean ball B with radius r centered at z, we define its circle-average-approximate-LQG measure by $M^\circ _{\gamma }(B) = r^{2+\gamma ^2/2} e^{\gamma h_r(z)}$, compare with (27). For $\delta >0$ and any two distinct points $u, v\in {{\mathbb {V}}}^\xi$, we define another variation of Liouville graph distance $D^{\circ }_{\gamma , \delta , \xi }(u, v)$ to be the minimal d such that there exist Euclidean balls $B_1, \ldots , B_d \subseteq {\mathbb {V}}^\xi$ with rational centers and $M^\circ _\gamma (B_i)\le \delta ^2$ for $1\le i\le d$, whose union contains a path from u to v.

We define $D'_{\gamma , \delta , \xi }(x, y)$ to be a version of the approximate Liouville graph distance where we restrict to cells in ${\mathbb {V}}^\xi$. One can verify that our proofs for Lemmas 3.5, 3.8, 3.10 and Corollary 3.9 as well as Proposition 3.17 extend automatically to $D'_{\gamma , \delta , \xi }$. Recall $C_{\mathrm {Mc}}$ as specified in Lemma 3.1.

Proposition 6.2

For every fixed $0<\xi < C_{\mathrm {Mc}}/3$ there exists a constant $c=c(\gamma , \xi )$ so that for every fixed $\iota >0$ and every sequence of $\xi$-admissible pairs $(A_\delta , B_\delta )$,

$$\begin{aligned} \min _{x\in A_\delta , \ y \in B_\delta }D_{\gamma , \delta }(x, y) \cdot \delta ^{\iota } \le \min _{x\in A_\delta , \ y\in B_\delta }D^{({\textsf {2} })}_{\gamma , \delta , \xi }(x, y) \le \min _{x\in A_\delta , \ y\in B_\delta } D_{\gamma , \delta }(x, y) \cdot \delta ^{-\iota }, \end{aligned}$$

with $(c\cdot \iota ^2)$-high probability. The preceding statement remains true if we replace $D^{({\textsf {2} })}_{\gamma , \delta , \xi }$ by $D^{\circ }_{\gamma , \delta , \xi }$.

Proof

By Lemma 6.1 and Proposition 3.17, we have that with $(c\cdot \iota ^2)$-high probability

$$\begin{aligned} \min _{x\in A_\delta , \ y\in B_\delta }D'_{\gamma , \delta }(x, y) \cdot \delta ^{\iota } \le \min _{x\in A_\delta , \ y \in B_\delta }D'_{\gamma , \delta , \xi }(x, y) \le \min _{x\in A_\delta , \ y \in B_\delta } D'_{\gamma , \delta }(x, y) \cdot \delta ^{-\iota }. \end{aligned}$$

Combined with Proposition 3.2, it implies that Proposition 6.2 follows provided that with $(c\cdot \iota ^2)$-high probability

$$\begin{aligned} \begin{aligned}&\min _{x\in A_\delta , \ y\in B_\delta }D'_{\gamma , \delta , \xi }(x, y) \cdot \delta ^{\iota } \le \min _{x\in A_\delta , \ y\in B_\delta }D^{({\textsf {2} })}_{\gamma , \delta , \xi }(x, y) \le \min _{x\in A_\delta , \ y\in B_\delta } D'_{\gamma , \delta , \xi }(x, y) \cdot \delta ^{-\iota },\\&\min _{x\in A_\delta , \ y\in B_\delta }D'_{\gamma , \delta , \xi }(x, y) \cdot \delta ^{\iota } \le \min _{x\in A_\delta , \ y \in B_\delta }D^{\circ }_{\gamma , \delta , \xi }(x, y) \le \min _{x\in A_\delta , \ y \in B_\delta } D'_{\gamma , \delta , \xi }(x, y) \cdot \delta ^{-\iota }. \end{aligned}\nonumber \\ \end{aligned}$$

(139)

The proof of (139) is similar to that of Proposition 3.2. Thus, we only briefly discuss how to adapt the proof of Proposition 3.2.

For $D^{({\textsf {2} })}_{\gamma , \delta , \xi }$, since $D^{({\textsf {2} })}_{\gamma , \delta , \xi } \ge D_{\gamma , \delta , \xi }$, it remains to bound $D^{({\textsf {2} })}_{\gamma , \delta , \xi }$ by $D'_{\gamma , \delta , \xi }$ from above. We repeat the proof of Proposition 3.2, but with the following change: we will now define a new version of $\Phi _{B, \delta }$ (similar to that in Definition 3.6) to be the minimal number of Euclidean balls ${\textsf {B} }$ with $M_\gamma (2{\textsf {B} }) \le \delta ^2$ that covers $\partial B$. (The only difference is that we used $M_{\gamma }(2{\textsf {B} })$ in the preceding definition as opposed to $M_{\gamma }({\textsf {B} })$ as in Definition 3.6.) One can then just repeat the arguments with this version of $\Phi _{B, \delta }$ to conclude the proof on the upper bound—the only place that needs to be changed is in the proof of (50) and (51), where the required change is noting but enlarging a few constants which have been absorbed by much larger terms in the earlier proof.

Next, we consider $D^{\circ }_{\gamma , \delta , \xi }$. By [14, Proposition 3.2] (which states that the circle average process and our ${\hat{h}}$-process are close to each other) and Lemma 2.8, we get that with high probability

$$\begin{aligned} \max _{j: 2^{-j} \ge \delta ^{C_{\mathrm {mc}}+10}} \ \max _{x\in {\mathbb {V}}^\xi } \left| \eta _{2^{-j}}(x) - h_{2^{-j}}(x) \right| = O \left( \sqrt{\log \delta ^{-1}} \right) . \end{aligned}$$

This, together with Lemma 3.4, implies that with high probability

$$\begin{aligned} \min _{x\in A_\delta , \ y\in B_\delta }D'_{\gamma , \delta e^{(\log \delta ^{-1})^{0.6}}, \xi }(x, y)\le & {} \min _{x\in A_\delta , \ y\in B_\delta }D^{\circ }_{\gamma , \delta , \xi }(x, y) \\\le & {} \min _{x\in A_\delta , \ y\in B_\delta } D'_{\gamma , \delta e^{-(\log \delta ^{-1})^{0.6}}, \xi }(x, y). \end{aligned}$$

Combining Lemma 3.5, we complete the proof of (139), and thus the proof of the proposition. $\quad \square$