Baxter permuton and Liouville quantum gravity

The Baxter permuton $\mu_{B}$ is a random probability measure on the unit square (with uniform marginals). Our first result is an explicit expression of its intensity measure, defined by $\mathbb{E}[\mu_{B}](\cdot)\coloneqq\mathbb{E}[\mu_{B}(\cdot)]$, which answers [DP14, Question 6.7].

We highlight that the intensity measure of other universal random limiting permutons has been investigated in the literature. For instance, the intensity measure of the biased Brownian separable permuton, was determined by Maazoun in [Maa20]. We recall that the biased Brownian separable permuton $\mu^{q}_{S}$, defined for all $q\in[0,1]$, is a one-parameter universal family of limiting permutons arising form pattern-avoiding permutations (see Section 1.2 for more explanations). In [Maa20, Theorem 1.7], it was proved that for all $q\in(0,1)$, the intensity measure $\mathbb{E}[\mu^{q}_{S}]$ of the biased Brownian separable permuton is absolutely continuous with respect to the Lebesgue measure on $[0,1]^{2}$. Furthermore, $\mathbb{E}[\mu^{q}_{S}]$ has the following density function

The proof of [Maa20, Theorem 1.7] relies on an explicit construction of the biased Brownian separable permuton $\mu^{q}_{S}$ from a one-dimensional Brownian excursion decorated with i.i.d. plus and minus signs. To the best of our knowledge this proof cannot be easily extended to the Baxter permuton case. See the figures below for some plots of $p_{B}(x,y)$ and $p^{q}_{S}(x,y)$ using numerical approximations of the integrals.

Our second result is Theorem 1.7 states that the Baxter permuton a.s. contains a positive density of every possible (standard) pattern. To state our result, we first define the permutation induced by $k$ points in the square $[0,1]^{2}$. Recall that $[n]=\left\{1,\dots,n\right\}$.

We now define the random permutation of size $k$ induced by a deterministic permuton.

We will also consider random permutations induced by random permutons $\mu$. In order to do that, we need to construct a sequence $(X_{i},Y_{i})_{i\in[k]}$, where the points $(X_{i},Y_{i})$ are independent with common distribution $\mu$ conditionally on $\mu$. This is possible up to considering a new probability space where the joint distribution of $(\mu,(X_{i},Y_{i})_{i\in[k]})$ is determined as follows: for every positive measurable functional $H:\mathcal{M}\times[0,1]^{2k}\to\mathbb{R}$,

We now recall some standard notation related to permutation patterns; see Appendix A for more details. Let $\mathcal{S}_{n}$ be the set of permutations of size $n$ and $\mathcal{S}=\bigcup_{n\in\mathbbm{Z}_{>0}}\mathcal{S}_{n}$ be the set of permutations of finite size. Fix $k\leq n$ and $\sigma\in\mathcal{S}_{n}$. Given a subset $I$ of cardinality $k$ of the indices of $\sigma$, the pattern induced by $I$ in $\sigma$, denoted $\operatorname{pat}_{I}(\sigma)$, is the permutation corresponding to the diagram obtained by rescaling the points $(i,\sigma(i))_{i\in I}$ in a $|I|\times|I|$ table (keeping the relative position of the points). If $\operatorname{pat}_{I}(\sigma)=\pi\in\mathcal{S}_{k}$ we will say that $(\sigma(i))_{i\in I}$ is an occurrence of $\pi$ in $\sigma$. We denote by $\operatorname{occ}(\pi,\sigma)$ the number of occurrences of a pattern $\pi$ in a permutation $\sigma$. Moreover, we denote by $\widetilde{\operatorname{occ}}(\pi,\sigma)$ the proportion of occurrences of $\pi$ in $\sigma,$ that is,

We finally recall an important fact about permuton convergence. Suppose $(\sigma_{n})_{n}$ is a sequence of random permutations converging in distribution in the permuton sense to a limiting random permuton $\mu$, i.e. $\mu_{\sigma_{n}}\xrightarrow[]{d}\mu$. Then, from [BBF⁺20, Theorem 2.5], it holds that $\left(\widetilde{\operatorname{occ}}(\pi,\sigma_{n})\right)_{\pi\in\mathcal{S}}$ converges in distribution in the product topology as $n\to\infty$ to the random vector $\left(\widetilde{\operatorname{occ}}(\pi,\mu)\right)_{\pi\in\mathcal{S}}$, where the random variables $\widetilde{\operatorname{occ}}(\pi,\mu)$ are defined for all $\pi\in\mathcal{S}$ as follows

Our result is quenched in the sense that for almost every realization of the Baxter permuton $\mu_{B}$, it contains a strictly positive proportion of every pattern $\pi\in\mathcal{S}$. Since pattern densities of random permutations converge to pattern densities of the corresponding limiting random permuton, we have the following corollary of Theorems 1.2 and 1.7.

Fix $\gamma\in(0,2)$ and some angle $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$. In what follows we consider:

• •

  a unit-area $\gamma$-Liouville quantum sphere $(\widehat{{\mathbb{C}}},h,0,\infty)$ with two marked points at 0 and $\infty$ and associated $\gamma$-LQG area measure $\mu_{h}$ (see Definition 2.1);

• •

  an independent whole-plane GFF $\widehat{h}$ (see Section 2.1.1);

• •

  two space-filling SLE${}_{\kappa^{\prime}}$ counterflow lines of $\widehat{h}$ in $\widehat{{\mathbb{C}}}$ with angle $0$ and ${\theta-\frac{\pi}{2}}$ constructed from angle $\frac{\pi}{2}$ and $\theta$ flow lines with $\kappa^{\prime}=16/\gamma^{2}$ (see Section 2.1.3). We denote these two space-filling SLE${}_{\kappa^{\prime}}$ curves from $\infty$ to $\infty$ by $\eta^{\prime}_{0}$ and $\eta^{\prime}_{\theta-\frac{\pi}{2}}$.

We emphasize the independence of the counterflow lines and the quantum sphere. In addition, we assume that the curves $\eta^{\prime}_{0}$ and $\eta^{\prime}_{\theta-\frac{\pi}{2}}$ are parametrized so that $\eta^{\prime}_{0}(0)=\eta^{\prime}_{0}(1)=\eta^{\prime}_{\theta-\frac{\pi}{2}}(0)=\eta^{\prime}_{\theta-\frac{\pi}{2}}(1)=\infty$ and $\mu_{h}(\eta^{\prime}_{0}([s,t]))=\mu_{h}(\eta^{\prime}_{\theta-\frac{\pi}{2}}([s,t]))=t-s$ for $0\leq s<t\leq 1$. We have the following result.

For every fixed $\gamma\in(0,2)$, the function

is a decreasing homeomorphism and therefore has an inverse function $\theta_{\gamma}(q)$. Finally, for all $\theta\in[0,\pi/2]$ and all $\gamma\in(0,2)$, it holds that $q_{\gamma}(\theta)+q_{\gamma}(-\theta)=1$. In particular, $q_{\gamma}(0)=1/2$ for all $\gamma\in(0,2)$. The Baxter permuton corresponds to $\gamma=4/3$ and $\theta=0$.

To prove Theorem 1.3, we first extend Theorem 1.12 to give a more explicit description of the skew Brownian permuton measure $\mu_{\rho,q}$ in terms of a unit-area quantum sphere and two space-filling SLEs, i.e. we prove the following.

Using Proposition 1.13, we express the intensity measure $\mathbb{E}[\mu_{\rho,q}]$ in terms of a quantum sphere decorated by certain flow lines of a Gaussian free field, which are simple SLE_κ curves with $\kappa=\gamma^{2}\in(0,4)$. This is proved in Proposition 3.7 using the rerooting invariance of marked points for quantum spheres ([DMS21]; see also Proposition 3.6 below) and the fact that the outer boundaries of the SLE_16/κ-type curves $\eta^{\prime}_{0}$ and $\eta_{\theta-\frac{\pi}{2}}^{\prime}$ are simple SLE_κ curves. Using conformal welding results and scaling properties of quantum disks and spheres ([AHS20]; see also Section 3.1), this leads to a simpler expression for $\mathbb{E}[\mu_{\rho,q}]$ via the density function $p_{W}(a,\ell_{1},\ell_{2})$ of the area $a$ of quantum disks with given quantum boundary lengths $\ell_{1}$ and $\ell_{2}$ (see Theorem 3.8). When $(\rho,q)=(-1/2,1/2)$, the density $p_{W}(a,\ell_{1},\ell_{2})$ is the same as the density of the duration of a Brownian excursion in a cone of angle $\frac{\pi}{3}$ (as argued in [AHS20, Section 7]; see also Proposition 3.3). The latter can be computed using standard heat equation argument as done in Section 3.3 via [Iye85]. We finally briefly explain the relation between the integrals in (1.2) and spherical tetrahedra (see Section 3.3.3).

To prove Theorem 1.10, we begin by observing that the occurrence of a fixed (standard) pattern in $\mu_{\rho,q}$ can be reformulated in terms of a specific condition on the crossing and merging order of some collection of flow lines of a Gaussian free field (see Lemma 4.1 and Figure 7 for a precise statement). Then building on the key result from [MS17, Lemma 3.8], which roughly speaking states that a simple SLE_κ curve can approximate any continuous simple curves with positive probability, we prove that this crossing and merging condition holds with positive probability (the main difficulty here is that we need to look at several flow lines of different angles together). Finally, by the scaling invariance of the whole-plane GFF and a tail triviality argument, we conclude the proof of Theorem 1.10.

We conclude the introduction with three observations. First, qualitatively, our method works equally well for $\mu_{\rho,q}$ with any $\rho\neq 1$. But quantitatively, the Baxter permuton corresponds to a special case where the function $p_{W}(a,\ell_{1},\ell_{2})$ is significantly simpler than the general case; see Remark 3.14. This is why Theorem 1.3 is restricted to the Baxter case while Theorem 1.10 is for the general case.

Second, the angle $\theta$ in Theorem 1.12 has a simple permutation interpretation.

The third and fourth named authors of this paper are working on deriving an exact formula for $q_{\gamma}(\theta)$ with Ang. For the Baxter permuton $\mathbb{E}[(\widetilde{\operatorname{occ}}(21,\mu_{\rho,q}))]=\frac{1}{2}$ by symmetry, which is consistent with $\theta=0$. In fact, we can express $\mathbb{E}[(\widetilde{\operatorname{occ}}(\pi,\mu_{\rho,q}))]$ for any pattern $\pi$ in terms of SLE and LQG, but for $\pi\neq 21$ we do not find any exact information as enlightening as (1.5); see the end of Section 3. Note that Proposition 1.14 answers [Bor21b, Conjecture 1.22], which conjectures that the expected proportion of inversions in $\mu_{\rho,q}$ is an increasing function of $q$ (recall that $\theta$ is a decreasing function of $q$).

Finally, our work falls into the line of work proving integrability results (i.e., exact formulas) via SLE and/or LQG techniques. Integrability results for SLE and LQG can be proven via a variety of methods, see e.g. [Sch01, Law05a, Dub06, GTF06, SSW09, SZ10, SW11, HS11, BI12, AKL12, BJV13, LV19, ALS22] for SLE results and [KRV20, Rem20] for LQG results. As a particular class of methods, couplings between SLE and LQG [She16, DMS21] allows to exploit the interplay between SLE and LQG to prove new results about both objects. For example, the KPZ formula [KPZ88] has been used to predict exponents of statistical mechanics models and dimensions of SLE curves via combinatorics on planar maps; see e.g. [GHM20] for rigorous computations in this spirit. Chen, Curien and Maillard [CCM20] give a heuristic derivation of the conformal radius formula in [SSW09] by using the coupling with LQG; see also [HL22] for a proof via LQG techniques. A number of integrability results based on the SLE and LQG coupling have been established by subsets of the coauthors of this paper [AHS20, AHS21, ARS21, AS21], and our Theorem 1.3 can be viewed as a part of this ongoing endeavor.

Recall that the Gaussian free field (GFF) with free boundary conditions (resp. zero boundary conditions) $h$ on a planar domain $D\subsetneq\mathbb{C}$ is defined by taking an orthonormal basis $\{f_{n}\}$ of $H(D)$ (resp. $H_{0}(D)$), the Hilbert space completion of the set of smooth functions on $D$ with finite Dirichlet energy (resp. finite Dirichlet energy and compact support) with respect to the Dirichlet inner product, an i.i.d. sequence $\{\alpha_{n}\}$ of standard normal random variables, and considering the sum $h=\sum_{n=1}^{\infty}\alpha_{n}f_{n}$. This series converges in an appropriate Sobolev space and hence in the space of distributions; see for instance [BP21, Theorem 1.24]. In the case of the free boundary GFF, we view $h$ as a distribution modulo a global additive constant (see also [BP21, Definition 5.2]).

The whole-plane Gaussian free field $h$, viewed as a distribution on $\mathbb{C}$ modulo a global additive constant, is defined in a similar manner as the free boundary GFF, but with $D=\mathbb{C}$ (see for instance [BP21, Section 5.4]). Sometimes we will fix this additive constant or view the whole-plane GFF as a distribution modulo a global additive integer multiple of some other fixed constant; if this is done, it will be always specified in the paper. We refer to [She07, WP20, BP21] for more background on the GFF.

Consider the space of pairs $(D,h)$, where $D\subseteq\mathbb{C}$ is a planar domain and $h$ is a distribution on $D$ (often some variant of the GFF). Define the equivalence relation $\sim_{\gamma}$, where $(D,h)\sim_{\gamma}(\widetilde{D},\widetilde{h})$ if there is a conformal map $\varphi:\widetilde{D}\to D$ such that

A quantum surface $S$ is an equivalence class of pairs $(D,h)$ under the relation $\sim_{\gamma}$, and we say $(D,h)$ is an embedding of $S$ if $S=(D,h)/\mathord{\sim}_{\gamma}$. In this paper, the domain $D$ shall be either the upper half plane $\mathbb{H}:=\{z\in\mathbb{C}:\mathrm{Im}\ z>0\}$, the Riemann sphere $\widehat{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}$, or a planar domain cut out by $\mathrm{SLE}_{\kappa}$ curves. We will often abuse notation and identify the pair $(D,h)$ with its equivalence class, e.g. we may refer to $(D,h)$ as a quantum surface (rather than a representative of a quantum surface).

A quantum surface with $k$ marked points is an equivalence class of elements of the form $(D,h,x_{1},\dots,x_{k})$, where $(D,h)$ is a quantum surface, the points $x_{i}\in{\overline{D}}$, and with the further requirement that marked points (and their ordering) are preserved by the conformal map $\varphi$ in (2.1).

A curve-decorated quantum surface is an equivalence class of tuples $(D,h,\eta_{1},...,\eta_{k})$, where $(D,h)$ is a quantum surface, $\eta_{1},...,\eta_{k}$ are curves in $\overline{D}$, and with the further requirement that $\eta$ is preserved by the conformal map $\varphi$ in (2.1). Similarly, we can define a curve-decorated quantum surface with $k$ marked points. Throughout this paper, the curves $\eta_{1},...,\eta_{k}$ are $\mathrm{SLE}_{\kappa}$ type curves (which have conformal invariance properties) sampled independently of the surface $(D,h)$.

Given some variant $h$ of the GFF, one can make sense of the bulk measure $\mu_{h}$, where $\mu_{h}(A)=\int_{A}e^{\gamma h(z)}\,dz$, by considering the circle average $h_{\varepsilon}(z)$ of $h$ on the circle $\partial B_{\varepsilon}(z)$ and taking the weak limit of $\varepsilon^{\gamma^{2}/2}e^{\gamma h_{\varepsilon}(z)}dz$ as $\varepsilon\to 0$; see for instance [Kah85, RV11, DS11]. The measure $\mu_{h}$ is then called the $\gamma$-Liouville quantum gravity area measure. Similarly, one can define a length measure $\nu_{h}$, called the $\gamma$-Liouville quantum gravity length measure, on certain curves in the closure $\overline{D}$ of $D$. Notice that $\mu_{h}$ and $\nu_{h}$ depend on the parameter $\gamma$, but we skip $\gamma$ from the notation since the value of $\gamma$ will always be implicitly understood from the context. The two measures satisfy natural scaling properties, namely $\mu_{h+c}(A)=e^{\gamma c}\mu(A)$ and $\nu_{h+c}(S)=e^{\frac{\gamma}{2}c}\nu(S)$ for $c\in\mathbbm{R}$ an arbitrary constant. See [DS11] for more details on these bulk and boundary Liouville quantum gravity measures. A self-contained introduction to GFF, Liouville quantum gravity measures, and quantum surfaces, can be also found in [GHS19, Section 3.2-3] or in the lecture notes [BP21].

Now we formally introduce quantum spheres and quantum disks, which are the main types of quantum surfaces considered in this paper and are defined in terms of some natural variants of the GFF. The reader can find intuitive explanations of the next definitions at the end of this section. We also highlight that it is not strictly necessary to understand the technical details involved in the following definitions in order to then follow our proofs in the consecutive sections.

As argued in [DMS21, Section 4.1], when $D=\widehat{\mathbb{C}}$ (resp. $D=\mathbb{H}$), we have the decomposition $H(\widehat{\mathbb{C}})=H_{1}(\widehat{\mathbb{C}})\oplus H_{2}(\widehat{\mathbb{C}})$ (resp.  $H(\mathbb{H})=H_{1}(\mathbb{H})\oplus H_{2}(\mathbb{H})$), where $H_{1}(\widehat{\mathbb{C}})$ (resp. $H_{1}(\mathbb{H})$) is the subspace of radially symmetric functions, and $H_{2}(\widehat{\mathbb{C}})$ (resp. $H_{2}(\mathbb{H})$) is the subspace of functions having mean 0 on all circles $\{|z|=r\}$ (resp. semicircles $\{|z|=r,\ \text{Im}\ z>0\}$). For a whole-plane GFF $h$, we can decompose $h=h_{1}+h_{2}$, where $h_{1}$ and $h_{2}$ are independent distributions given by the projection of $h$ onto $H_{1}(\widehat{\mathbb{C}})$ and $H_{2}(\widehat{\mathbb{C}})$, respectively. We remark that $h_{1}$ is defined modulo an additive constant while $h_{2}$ is not. The same result applies for the upper half plane $\mathbb{H}$.

Since a quantum surface is an equivalence class of pairs $(D,\psi)$ (or, more generally, an equivalence class of tuples $(D,\psi,z_{1},\dots,z_{k})$ with $z_{1},\dots,z_{k}\in\overline{D}$), in order to describe the law of a quantum sphere, we will start by describing the law of its random field $\psi$.

It is explained in [DMS21, Sections 4.2 and 4.5] that the considered conditioning on $B$ and $\widetilde{B}$, along with the conditioning on the quantum area of a weight $4-\gamma^{2}$ quantum sphere, can be made rigorous via a limiting procedure, although we are conditioning on probability zero events.

We remark that the weight $4-\gamma^{2}$ here is “typical” because in this case the two marked points (which currently correspond to 0 and $\infty$) can be realized as independent samples from the $\gamma$-LQG area measure $\mu_{\psi}$ (see Proposition 3.6 below for a precise statement). This important rerooting invariance property shall later be used in Section 3.3 in order to compute the density of the Baxter permuton via quantum surfaces.

We now turn to the definition of quantum disks, which is splitted in two different cases: thick quantum disks and thin quantum disks.