Metric Growth Dynamics in Liouville Quantum Gravity

Abstract

We consider the metric growth in Liouville quantum gravity (LQG) for $\gamma \in (0,2)$. We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary, for every $\gamma \in (0,2)$. The infinitesimal version of this stationarity combined with an explicit expression of the generator of the evolution of the trace field $(h_t)$ provides a formal invariance equation that a measure on trace fields must satisfy. When considering a modified process corresponding to an evolution of LQG surfaces, we prove that the invariance equation is satisfied by an explicit $\sigma$-finite measure on trace fields. This explicit measure on trace fields only corresponds to the pure gravity case. On the way to prove this invariance, we retrieve the specificity of both $\gamma = \sqrt{8/3}$ and of the LQG dimension $d_{\gamma } = 4$. In this case, we derive an explicit expression of the (nonsymmetric) Dirichlet form associated with the process $(h_t)$ and construct dynamics associated with its symmetric part.

Appendix

1.1 Gaussian identities

Gaussian integration by parts If $X,X_1,\dots ,X_n,Y$ are variables in a centered Gaussian space, $\psi \in C^\infty _c$, then by integration by parts,

$$\begin{aligned} {\mathbb {E}}(\psi '(X)) {{\,\textrm{Var}\,}}(X)={\mathbb {E}}(X\psi (X)), \quad {\mathbb {E}}(\psi '(X)) {{\,\textrm{Cov}\,}}(X,Y)={\mathbb {E}}(\psi (X)Y) \end{aligned}$$

(6.1)

and more generally

$$\begin{aligned} {\mathbb {E}}(\psi (X_1,\dots ,X_n)Y)=\sum _i{\mathbb {E}}(\psi _i(X_1,\dots ,X_n)) {{\,\textrm{Cov}\,}}(X_i,Y). \end{aligned}$$

(6.2)

Cameron–Martin formula Recall first that if (X, Y) is a Gaussian vector and Y is centered, then for a smooth function F with some mild growth condition, ${\mathbb {E}}(F(X) e^{Y-{\mathbb {E}}(Y^2)/2}) = {\mathbb {E}}(F(X+{{\,\textrm{Cov}\,}}(X,Y)))$. This is referred to as the Cameron–Martin formula and can be interpreted by a shift of the mean of the distribution of X when the reference measure is tilted by $e^{Y-{\mathbb {E}}(Y^2)/2}$. We gather here some consequences of this identity.

Let (X, Y, W, Z) be a Gaussian vector with (X, Y) being centered. By applying the Cameron–Martin formula with shift Y, we have

$$\begin{aligned} {\mathbb {E}}( W e^{Y- {\mathbb {E}}(Y^2)/2 }) = {\mathbb {E}}(W Y). \end{aligned}$$

(6.3)

With the shift $X+Y$, we have

$$\begin{aligned}{} & {} {\mathbb {E}}(W Z e^{X-{\mathbb {E}}(X^2)/2} e^{Y-{\mathbb {E}}(Y^2)/2}) \nonumber \\{} & {} \quad = \left( {\mathbb {E}}(WZ) + ({\mathbb {E}}(WX)+{\mathbb {E}}(WY))({\mathbb {E}}(ZX)+{\mathbb {E}}(ZY)) \right) e^{{\mathbb {E}}(XY)} \end{aligned}$$

(6.4)

and

$$\begin{aligned}{} & {} {\mathbb {E}}\left( \left( W -{\mathbb {E}}(W e^{X-{\mathbb {E}}(X^2)/2}) \right) \left( Z -{\mathbb {E}}(Z e^{Y-{\mathbb {E}}(Y^2)/2}) \right) e^{X-{\mathbb {E}}(X^2)/2} e^{Y-{\mathbb {E}}(Y^2)/2} \right) \nonumber \\{} & {} \quad = \left( {\mathbb {E}}(WY) {\mathbb {E}}(ZX) + {\mathbb {E}}(WZ) \right) e^{{\mathbb {E}}(X Y)}. \end{aligned}$$

(6.5)

Alternative proofs using the Cameron–Martin formula Several computations carried out could have been done using the Cameron–Martin formula (2.10). Since this way of computing is sometimes useful, but not in the spirit of most of the computations carried above, we present here some applications of this formula where we reinterpret some of our results.

Lemma 22

Consider a covariance kernel given on $\mathbb {U}^2$ by $G_{\partial }(w,z) = -2 \log |w-z|$ and a bulk test functional $F = \varphi (\int h p_1 d\lambda , \dots , \int h p_n d\lambda )$. If $f(h,w) := \widetilde{DF}(h,w)$ then $f(h-\xi G_{\partial }(\cdot , w)) = \frac{1}{2\pi \xi } \partial _\theta F (h-\xi G_{\partial }(\cdot ,w))$.

Proof

First, note that $\widetilde{\partial _n H p} = \partial _\theta p$: if $p = \sum \alpha _k e_k$, then $\widetilde{\partial _n H p} = - \sum \alpha _k \lambda _k \widetilde{e}_k = \sum \alpha _k \partial _\theta e_k = \partial _\theta p$. When $f(h,x) = \sum _i \partial _i \varphi ( \int h p_1 d\lambda , \dots , \int h p_n d\lambda ) {\tilde{p}}_i(x)$, introduce $P_i$ s.t. $\partial _n H P_i(x) = p_i(x)$ so that

$$\begin{aligned} \int G_{\partial }(\cdot , x) p_i(\cdot ) d\lambda = \int G_{\partial }(\cdot , x) \partial _n H P_i(\cdot ) d\lambda = \int \partial _n H G_{\partial }(\cdot , x) P_i(\cdot ) d\lambda = -2\pi P_i(x), \end{aligned}$$

where we used $\partial _n H G_{\partial }(\cdot ,x) = - 2\pi \delta _x(\cdot )$ to obtain the last equality. So, in this case, using ${\tilde{p}}_i(x) = \widetilde{\partial _n H P_i}(x) = \partial _\theta P_i(x)$ we have

$$\begin{aligned} f(h-\xi G_{\partial }(\cdot ,x),x)&= \sum _i \partial _i \varphi ( \int h p_1 d\lambda + 2\pi \xi P_1(x), \dots , \int h p_n d\lambda + 2\pi \xi P_n(x)) {\tilde{p}}_i(x) \\&= \frac{1}{2\pi \xi } \partial _\theta \varphi ( \int h p_1 d\lambda + 2\pi \xi P_1(x), \dots , \int h p_n d\lambda + 2\pi \xi P_n(x)) \end{aligned}$$

and this completes the proof. $\square$

Discussion on Lemma 13. We introduce and have

$$\begin{aligned} \mathcal {E}_c(F,G) := \int \langle {\widetilde{DF}} , {\widetilde{DG}} \rangle _{L^2(\mu )} d\rho = \int F (- \mathcal {L}_c G ) d \rho \end{aligned}$$

(6.6)

where

$$\begin{aligned} \mathcal {L}_c G = \sum _{i,j} \partial _{i,j} \psi \int {\tilde{q}}_i {\tilde{q}}_j d\mu - \frac{1}{2\pi \xi } \int \partial _n H DG d\mu . \end{aligned}$$

Then, Lemma 13 is equivalent to $\int \mathcal {L}_c F d\rho = 0$ which comes from (6.6) since then $\mathcal {L}_c^* = \mathcal {L}_c$ so $\int \mathcal {L}_c F d\rho = \int F \mathcal {L}^*1 d\rho = 0$. (6.6) is indeed a corollary of the Cameron–Martin formula and Lemma 22:

$$\begin{aligned} \int \langle {\widetilde{DF}} , {\widetilde{DG}} \rangle _{L^2(\mu )} d\rho= & {} \frac{1}{(2\pi \xi )^2} \int \langle \partial _\theta F_x, \partial _\theta G_x \rangle _{L^2(\lambda )} d\rho \\= & {} - \int \frac{1}{(2\pi \xi )^2} \langle F_x, \partial _\theta ^2 G_x \rangle _{L^2(\lambda )} d\rho \end{aligned}$$

and

$$\begin{aligned}{} & {} \partial _\theta G_x = 2\pi \xi \sum _i (\partial _i \psi )_x {\tilde{q}}_i(x), \\{} & {} \partial _\theta ^2 G_x = (2\pi \xi )^2 \sum _{i,j} (\partial _{i,j} \psi )_x {\tilde{q}}_i(x) {\tilde{q}}_j(x) + 2\pi \xi \sum _i (\partial _i \psi )_x \partial _\theta {\tilde{q}}_i(x) \end{aligned}$$

so, with $\partial _\theta {\tilde{q}}_i = - \partial _n H q_i$,

$$\begin{aligned} \int \langle {\widetilde{DF}} , {\widetilde{DG}} \rangle _{L^2(\mu )} d\rho = - \int F \left[ \sum _{i,j} \partial _{i,j} \psi \int {\tilde{q}}_i {\tilde{q}}_j d\mu - \frac{1}{2\pi \xi } \int \partial _n H DG d\mu \right] d\rho . \end{aligned}$$

Discussion on Lemma 21. For $F = F(h)$, we write $F_x$ for $F(h-\xi G_{\partial }(\cdot ,x))$. Assume $(p_1, \dots , p_n)$ is orthonormal in $L^2(\lambda )$. Then, for $F = \varphi (\int _{\mathbb {U}} h p_1 d\lambda , \dots , \int _{\mathbb {U}} h p_n d\lambda )$, $G = \psi (\int _{\mathbb {U}} h q_1 d\lambda , \dots , \int _{\mathbb {U}} h q_m d\lambda )$

$$\begin{aligned}&\int \langle DF, DG \rangle _{L^2(\mu )} d\rho \\&\quad = \iint _{\mathbb {U}} (DF)_x (DG)_x d\lambda d\rho = \sum _{i,j} \iint _{\mathbb {U}} (\partial _i \varphi )_x (\partial _j \psi )_x p_i(x) q_j(x) d\lambda d\rho \\&\quad = \sum _{i,j} \iint _{\mathbb {U}} D_{p_i} \varphi _x (\partial _j \psi )_x p_i(x) q_j(x) d\lambda d\rho \\&\quad = \sum _{i,j} \iint _{\mathbb {U}} \varphi _x \left[ - D_{p_i}(\partial _j \psi )_x - \frac{1}{2\pi } (\partial _j \psi )_x \int _{\mathbb {U}} ( \partial _n H h + \xi ) p_i d \lambda \right] p_i(x) q_j(x) d\lambda d\rho \\&\quad = \iint _{\mathbb {U}} \varphi _x \left[ - \sum _{j,j'} (\partial _{j,j'} \psi )_x \Pi _P(q_{j'})(x) q_j(x)\right. \\&\qquad \left. - \frac{\xi }{2\pi } \Pi _P(1)(x) (DG)_x - \frac{1}{2\pi } \Pi _P(\partial _n H h)(x) (DG)_x \right] d\lambda d\rho \end{aligned}$$

where the first equality uses the Cameron–Martin formula, the third one the fact that P is orthonormal and the fourth one an integration by parts.

1.2 Inverse mapping

We consider for $\xi \in (0,1)$ the GMC $e^{\xi h}$ on $\mathbb {U}$. This gives a coupling $(h, e^{\xi h})$ where $e^{\xi h}$ is measurable with respect to h. We want show the existence of a measurable map $I_{\xi }$ such that $\rho$-a.e., $h = I_{\xi }(e^{\xi h})$. The proof in [11] can be adapted for the log-correlated Gaussian field h on $\mathbb {U}$, in particular by relying on the detailed study of the field h and its chaos measures in [9]. We just sketch the main arguments below. First, set

$$\begin{aligned} B_{\varepsilon }(x) := \{ e^{i y} : y \in (x-\varepsilon ,x+\varepsilon ) \} \qquad \text {and} \qquad e^{\xi h^{\varepsilon }(e^{ix})}:= \mu _{\xi }(B_\varepsilon (x)) \end{aligned}$$

(6.7)

so $h^{\varepsilon }(x) = \xi ^{-1} \log \mu _{\xi }(B_\varepsilon (x))$. Then, the following convergence occurs in $L^2$

$$\begin{aligned} \int h p d\lambda = \lim _{\varepsilon \rightarrow 0} \int (h^{\varepsilon }(w) - {\mathbb {E}}h^{\varepsilon }(w) ) p(w) d\lambda \end{aligned}$$

To justify it, consider an approximation $h_{\varepsilon }(x)$ at space-scale (e.g., $h_{\varepsilon } := \langle h, \rho _{\varepsilon }^x \rangle$ for a mollification to be specified) and set $f_{\varepsilon }(x) := h^{\varepsilon }(x)-h_{\varepsilon }(x)$. For the convergence to occur, it is sufficient to have the following pointwise estimates: there exists $\alpha > 0$ and $0< \kappa < 1$ such that, uniformly in $\varepsilon \in (0,1/4)$,

$$\begin{aligned}&{{\,\textrm{Var}\,}}f_{\varepsilon }(x) \le C \log \varepsilon ^{-1} \end{aligned}$$

(6.8)

$$\begin{aligned}&| {{\,\textrm{Cov}\,}}f_{\varepsilon }(x), f_{\varepsilon }(y) | \le C \varepsilon ^{\alpha } \qquad \text {for } |x-y| > \varepsilon ^{\kappa } \end{aligned}$$

(6.9)

Indeed, by splitting $\mathbb {U}^2$ in points $\{|w-z| < \varepsilon ^{\kappa } \}$ and its complement, using the variance bound in the former case and the covariance one in the latter case, we get ${{\,\textrm{Var}\,}}\int p(e^{ix}) f_{\varepsilon }(x) dx \le C( \varepsilon ^{\kappa } \log \varepsilon ^{-1} + \varepsilon ^{\alpha } )$. The desired convergence follows then from the one of $\int _{\mathbb {U}} h_\varepsilon p d\lambda$.

We start with (6.8). Consider the GMC $\eta = e^{\xi \psi }$ associated with the field $\psi$ with covariance ${\mathbb {E}}(\psi (x) \psi (y)) = - \log (|x-y| \wedge 1)$ on $\mathbb {R}$. When fixing an interval I of length $< \pi$, by Lemma 3.6 and equation (53) of [9], there exists a coupling of $\mu$ and $\eta$ with a random variable X having Gaussian tails such that for any interval $B\subset I$, $e^{-X} \eta (B) \le \mu (B) \le \eta (B) e^X$. In particular, ${{\,\textrm{Var}\,}}\log \mu (B_{\varepsilon }(x)) \le 2 {\mathbb {E}}(X^2) + 2 {{\,\textrm{Var}\,}}\log \eta (B_{\varepsilon }(x))$ . The pointwise variance estimate (6.8) follows from the exact scaling relation (54) in [9] and from the upper bound on the pointwise variance of the mollification of a log-correlated field (which can be found in [10]) for ${{\,\textrm{Var}\,}}h_{\varepsilon }(x)$.

We sketch here the main ideas to get the covariance bound (6.9). Use a white-noise representation of the field to split it into two parts, a fine field whose restriction on $B_{\varepsilon }(x)$ and $B_{\varepsilon }(y)$ are independent and a coarse field, which is independent. This is useful to obtain the exact decorrelation of measure on these sets. In [9], the field h is represented with a white-noise on $\mathbb {U} \times \mathbb {R}^+$ (up to an independent additive constant) where the y-axis represents the spatial scale, so a natural way to consider a coarse field and a fine field is to split horizontally the domain of the white-noise ($\{ y < \varepsilon ^{\kappa } \}$ and its complement). Doing so, we write $h = h_{(0,\varepsilon ^{\kappa })} + h_{(\varepsilon ^{\kappa },\infty )}$. Then, note that the oscillation of the coarse field on a microscopic ball is of order ${{\,\textrm{osc}\,}}_{B_{\varepsilon }(x)} h_{(\varepsilon ^{\kappa },\infty )} = O(\varepsilon ^{1-\kappa })$ so that

$$\begin{aligned} \mu _h (B_{\varepsilon }(x)) = \mu _{h_{(0,\varepsilon ^{\kappa })}}(B_{\varepsilon }(x)) e^{\xi h_{(\varepsilon ^{\kappa },\infty )}(x)} e^{O(\varepsilon ^{1-\kappa })} \end{aligned}$$

and

$$\begin{aligned} \xi ^{-1} \log \mu _h (B_{\varepsilon }(x)) - h_{\varepsilon }(x)&= \xi ^{-1} \log \mu _{h_{(0,\varepsilon ^{\kappa })}} (B_{\varepsilon }(x)) + h_{(\varepsilon ^{\kappa },\infty )}(x) - h_{\varepsilon }(x) + O(\varepsilon ^{1-\kappa }) \\&= \left( \xi ^{-1} \log \mu _{h_{(0,\varepsilon ^{\kappa })}} (B_{\varepsilon }(x)) - h_{(\varepsilon ,\varepsilon ^{\kappa })}(x) \right) \\&\quad + \left( h_{(\varepsilon ,\infty )}(x) - h_{\varepsilon }(x) \right) + O(\varepsilon ^{1-\kappa }) \\&= M_x + F_x + R_x \end{aligned}$$

so that when $|x-y| > \varepsilon ^{\kappa }$, $M_x$ and $M_y$ are independent. We need pointwise variance upper bounds for $M_x$ and $F_x$. For $M_x$ this uses the same ideas as for the bound (6.8). For $F_x$, note that we have the decomposition in independent terms $h_{\varepsilon }(x) - h_{(\varepsilon ,\infty )}(x) = \langle h_{(0,\varepsilon )}, \rho _{\varepsilon }^x \rangle + \langle h_{(\varepsilon ,\infty )}, \rho _{\varepsilon }^x - \delta _x \rangle$. Now, with $\rho _{\varepsilon }^x(y) := \frac{1}{\pi } \sum _{k = 1}^{\varepsilon ^{-1}} \cos (k(x-y))$, $h_{(0,\varepsilon )}$ is essentially orthogonal in $L^2$ to $\rho _{\varepsilon }^x(y)$ and $\delta _x(y)-\rho _{\varepsilon }^x(y) = \frac{1}{\pi } \sum _{k > \varepsilon ^{-1}} \cos (k(x-y))$ to $h_{(\varepsilon ,\infty )}$ and one can get a polynomial upper bound on the variance.

1.3 Comparison with the QLE generator

The article [64] develops a formal SPDE satisfied by the QLE process, using the Brownian motions driving the SLEs. We introduce the notation used by the authors and translate their results in our notation to compare the generator. They work with a boundary probability measure $\nu _t$ instead of $\mu _t$ (a general Borel measure without mass constraint) and with a normalization of harmonic functions such that $h_t(0) = 0$. They argue that the dynamics of the harmonic functions $(h_t)$ are formally given by

$$\begin{aligned} {\dot{h}}_t(z) = \int _{\mathbb {U}} \left( D_t(z,u) + \mathcal {P}^{\star }(z,u) W(t,u) \right) d\nu _t(u) \end{aligned}$$

(6.10)

where

$$\begin{aligned} D_t(z,u)&= - \nabla h_t(z) \cdot \Phi (u,z) + \frac{1}{\sqrt{\kappa }} \mathcal {P}^{\star }(z,u) + Q (\partial _{\theta } \overline{\mathcal {P}})(z,u) \\&= - \nabla h_t(z) \cdot \Phi (u,z) + \xi \mathcal {P}^{\star }(z,u) + ( 2\xi + \frac{1}{2\xi }) (\partial _{\theta } \overline{\mathcal {P}})(z,u) \end{aligned}$$

where $\Phi (u,z) = -z \frac{z+u}{z-u}$, for $a,b \in \mathbb {C} = \mathbb {R}^2$, $a \cdot b = \Re ({\bar{a}} b),$ so $\nabla h_t(z) \cdot \Phi (u,z) = 2 \overline{\partial _{z} h_t} \cdot \Phi (u,z) = 2 \Re ( \Phi (u,z) \partial _z h_t )$ and

$$\begin{aligned} D_t(z,u) = - 2 \Re ( \Phi (u,z) \partial _z h_t ) + \xi \mathcal {P}^{\star }(z,u) + ( 2\xi + \frac{1}{2\xi }) (\partial _{\theta } \overline{\mathcal {P}})(z,u) \end{aligned}$$

(6.11)

Above $\mathcal {P}$ is $2\pi$ times the Poisson kernel on $\mathbb {D}$, i.e. $\mathcal {P}(z,w) = 2\pi H(z,w) = \Re \left( \frac{w+z}{w-z} \right)$, $\overline{\mathcal {P}}$ is $2\pi$ the conjugate Poisson kernel on $\mathbb {D}$, i.e. $\overline{\mathcal {P}}(z,w) = \Im \left( \frac{w+z}{w-z} \right)$ so $\partial _{\theta } \overline{\mathcal {P}}(z,w) = \Re \left( \frac{-2zw}{(w-z)^2} \right)$ and $\mathcal {P}^{\star } = \mathcal {P}-1$ so $\mathcal {P}^{\star }(0,w) = 0$ for all $w \in \mathbb {U}$. Finally, W is a space-time white noise on $\mathbb {U} \times [0,\infty )$. Furthermore, above we already specified $\xi = \frac{1}{\sqrt{\kappa }} = \frac{1}{\sqrt{6}}$ and $Q = (2\xi )^{-1}+2\xi = \frac{5}{\sqrt{6}}$.

We look at $d \int h_t f d\lambda$ for $f \in C_c^{\infty }(\mathbb {D})$. By Fubini, the part without white-noise contains three terms: for the first one, we integrate it over $d\nu$,

$$\begin{aligned} (i)= & {} - 2 \iint \Re ( \Phi (u,z) \partial _z h_t ) f(z) dz \nu (du)\\= & {} - 2 \iint h_t(z) \Re ( \partial _z \left( \Phi (u,z) f(z) \right) dz \nu (du) \\= & {} \int h_t(z) (D_{\nu }f)(z) dz \end{aligned}$$

using Fubini and recalling the expression (3.16) of $D_\nu f$ for the last equality.

using (4.13) for the last equality.

The part with white-noise becomes

For $F(h) = \int h f d\lambda = \int H h f d\lambda = \int h f^* d\lambda _{\partial }$, $DF = f^*$,

where

Now, we compute $\mathcal {L}F$ for $F = \varphi (\int h f_1 d\lambda , \dots , \int h f_n d\lambda )$. It is an application of Itô’s formula. The quadratic variation is given by

so, writing $p_i = f_i^*$,

$$\begin{aligned} \mathcal {L}F(h) = \sum _{i} b(p_i) \partial _i \varphi + \frac{1}{2} \sum _{i,j} \sigma (p_i, p_j) \partial _{i,j} \varphi \end{aligned}$$

where

and .