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The hyperbolic Brownian plane

Abstract

We introduce and study a new random surface, which we call the hyperbolic Brownian plane and which is the near-critical scaling limit of the hyperbolic triangulations constructed by Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The law of the hyperbolic Brownian plane is obtained after biasing the law of the Brownian plane of Curien and Le Gall (J Theoret Probab 27(4):1249–1291, 2014) by an explicit martingale depending on its perimeter and volume processes studied by Curien and Le Gall (Probab Theory Relat Fields 166(1):187–231, 2016). Although the hyperbolic Brownian plane has the same local properties as those of the Brownian plane, its large scale structure is much different since we prove e.g. that is has exponential volume growth.

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Notes

  1. We can find in the ball of radius r of \( \mathbf {T}_{\kappa }\) a number of points at distance at least \(\frac{r}{10}\) from each other that goes to \(+\infty \) as \(r \rightarrow +\infty \), so the sequence \(\big ( \frac{1}{r} B_r(\mathbf {T}_{\kappa }) \big )_{r \ge 1}\) is not tight for the Gromov–Hausdorff topology

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Acknowledgements

I thank Nicolas Curien for suggesting me to study this object, and for carefully reading many earlier versions of this manuscript. I also thank the anonymous referee for his useful comments. I acknowledge the support of ANR Liouville (ANR-15-CE40-0013) and ANR GRAAL (ANR-14-CE25-0014).

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Correspondence to Thomas Budzinski.

Appendix A: Proof of Proposition 10

Appendix A: Proof of Proposition 10

The goal of this appendix is to prove Proposition 10. We note that similar ideas to those below appear in Sect. 2 of [31], and more precisely in the proofs of Proposition 2.17 and 2.18. In particular, Lemma 30 is essentially proved in the proof of Proposition 2.18 there. If A is a subset of a metric space X and \(\varepsilon >0\), we will write \(A^{\varepsilon }\) for the union of all the open balls of radius \(\varepsilon \) centered at an element of A. Note that this is an open subset of X. We also recall that \(B_r(X)\) and \(\overline{B_r}(X)\) are respectively the closed ball and the hull centered at the root of X. In particular, they are both closed subsets of X. To prove Proposition 10, we will also need several times the following definition.

Definition 27

Let (Xd) be a metric space, \(x,y \in X\) and \(\varepsilon >0\). An \(\varepsilon \)-chain from x to y is a finite sequence of points \((z_i)_{0 \le i \le k}\) of X such that \(z_0=x\), \(z_k=y\) and \(d(z_i,z_{i+1}) \le \varepsilon \) for all \(0 \le i \le k-1\).

Lemma 28

Under the assumptions of Proposition 10, the application \(s \rightarrow \overline{B_s}(X)\) is continuous at r for the Hausdorff distance on the set of the compact subsets of \(\overline{B_{r+1}}(X)\).

Proof

Let \(\delta >0\). It is enough to show that, for some \(\varepsilon >0\), we have:

  1. (a)

    \(\overline{B_{r+\varepsilon }}(X) \subset \overline{B_r}(X)^{\delta }\),

  2. (b)

    \(\overline{B_r}(X) \subset \overline{B_{r-\varepsilon }}(X)^{\delta }\).

We start with the first point. Let \(A=\Big ( \bigcap _{\varepsilon >0} \overline{B_{r+\varepsilon }}(X) \Big ) \backslash \overline{B_r}(X)\). If \(x \in A\), then there is a geodesic from x to a point \(y \in X \backslash \overline{B_{r+1}}(X)\) that stays outside \(B_r(X)\). However, \(\gamma \) has to intersect \(B_{r+\varepsilon }(X)\) for all \(\varepsilon >0\). This is clearly impossible, so \(A=\emptyset \). In particular, the decreasing intersection of the compact sets \(\overline{B_{r+1/n}}(X) \backslash \overline{B_r}(X)^{\delta }\) is empty, so one of these compact sets is empty, which proves item (a).

For item b), let \(A'=\big ( \overline{B_r}(X)\backslash B_r(X) \big ) \backslash \Big ( \bigcup _{\varepsilon >0} \overline{B_{r-\varepsilon }}(X) \Big )\). By assumption (iii) we have \(\mu (A')=\emptyset \) and \(A'\) is open, so by assumption (ii) we get \(A'=\emptyset \). This implies \( \overline{B_r}(X)\backslash B_r(X) \subset \bigcup _{n \ge 0} \overline{B_{r-1/n}}(X)^{\delta }\). Moreover, we have \(B_r(X) \subset \overline{B_{r-1/n}}(X)^{\delta }\) for \(\frac{1}{n} < \delta \). Hence, the increasing family of open sets \(\overline{B_{r-1/n}}(X)^{\delta }\) covers the compact space \(\overline{B_r}(X)\), so there is an \(\varepsilon >0\) such that \(\overline{B_r}(X) \subset \overline{B_{r-\varepsilon }}(X)^{\delta }\).

Remark 29

Note that the first point is true even without assumptions (ii) and (iii).

Lemma 30

Let \(\varepsilon >0\). Let \((X, \rho , a)\) and \((Y, \rho , b)\) be two bipointed connected, compact subsets of a locally compact metric space (Zd). Assume that the Hausdorff distance between X and Y is less than \(\varepsilon \) and that \(d(a,b) \le \varepsilon \). Then, for all r such that \(r+4 \varepsilon < d(\rho ,b)\), we have

  1. 1)

    \(\overline{B_r}(X)^{\varepsilon } \cap Y \subset \overline{B_{r+4\varepsilon }}(Y)\),

  2. 2)

    \(\overline{B_r}(X) \subset \overline{B_{r+4\varepsilon }}(Y)^{\varepsilon }\).

Fig. 1
figure 1

Illustration of the proof of Lemma 30

Proof

We first notice that the connectedness of X implies that, for any two points x and \(x'\) in X, there is an \(\varepsilon \)-chain in X from x to \(x'\). The same is true for Y.

Let y be a point in \(\overline{B_r}(X)^{\varepsilon } \cap Y\). We want to show that \(y \in \overline{B_{r+4\varepsilon }}(Y)\). We can assume \(d(\rho , y)>r+4\varepsilon \) (if it is not the case, then \(y \in \overline{B_{r+4\varepsilon }}(Y)\) is obvious). Let \((z_i)_{0 \le i \le k}\) be an \(\varepsilon \)-chain from y to b in Y. We know that \(d(a,b) \le \varepsilon \) and that there is a point \(x \in X\) such that \(d(x,y)\le \varepsilon \). We write \(w_0=x\), \(w_k=a\) and, for all \(1 \le i \le k-1\), we take \(w_i \in X\) such that \(d(w_i,z_i) \le \varepsilon \) (see Fig. 1 for an illustration). For all i, we have \(d(w_i,w_{i+1}) \le d(w_i,z_i)+d(z_i,z_{i+1})+d(z_{i+1},w_{i+1}) \le 3 \varepsilon \), so \((w_i)\) is a \(3\varepsilon \)-chain from x to a in X. Since \(x \in \overline{B_r}(X)\), there must be at least one i such that \(d(\rho ,w_i) \le r+3\varepsilon \), which implies \(d(\rho ,z_i) \le r+4\varepsilon \). Hence, every \(\varepsilon \)-chain from y to b has at least one point at distance from the root less than \(r+4\varepsilon \), so there is no \(\varepsilon \)-chain from y to b in \(Y \backslash B_{r+4 \varepsilon }(Y)\). This implies that y and b do not lie in the same connected component of \(Y \backslash B_{r+4 \varepsilon }(Y)\), so \(y \in \overline{B_{r+4\varepsilon }}(Y)\), which proves the first point of the lemma.

The second point is easily obtained from the first one: if \(x \in \overline{B_r}(X)\), then there is a \(y \in Y\) such that \(d(x,y) \le \varepsilon \). By the first point, we have \(y \in \overline{B_{r+4\varepsilon }}(Y)\), so \(x \in \overline{B_{r+4\varepsilon }}(Y)^{\varepsilon }\). \(\square \)

Proof of Proposition 10

First, we need to prove that the radii of the \(\overline{B_r}(X_n)\) are bounded. Let \(R=\max \{ d(\rho ,x) | x \in \overline{B_{r+4}}(X)\}\) be the radius of \(\overline{B_{r+4}}(X)\). For n large enough, we have

$$\begin{aligned} d_{GH}(B_{R+3}(X_n),B_{R+3}(X))<1. \end{aligned}$$

We write \(X'=B_{R+3}(X)\) and \(X'_n=B_{R+3}(X_n)\). The above inequality means that we can embed \(X'\) and \(X'_n\) isometrically in the same space (Zd) in such a way that \(X'_n \subset (X')^1\) and \(\rho _n=\rho \).

Let \(b \in X'_n \backslash B_{R+2}(X_n)\) and \(a \in X'\) such that \(d(a,b) \le 1\). We have \(d(a,\rho ) \ge R+1\), so \(a \notin \overline{B_r}(X)\) and \(\overline{B_r}(X)=\overline{B_r}(X')\) is the hull of radius r with center \(\rho \) with respect to a in \(X'\). By item 2) of Lemma 30 for \(\varepsilon =1\), we have \(\overline{B_r}(X'_n) \subset \overline{B_{r+4}}(X')^1\). In particular, the radius of \(\overline{B_r}(X'_n)\) is less than \(R+1\). This means that only one connected component of \(X' \backslash B_r(X'_n)\) contains points at distance greater than \(R+1\) from the root. In other words, the radius of \(\overline{B_r}(X_n)\) is at most \(R+1\).

We now move on to the proof of our proposition. Let \(\delta >0\). By Lemma 28, there is \(\varepsilon >0\) such that \(\overline{B_{r+4 \varepsilon }}(X) \subset \overline{B_r}(X)^{\delta }\) and \(\overline{B_r}(X) \subset \overline{B_{r-4 \varepsilon }}(X)^{\delta }\). For n large enough, we have \(d_{GHP}(X',X'_n) \le \varepsilon \). This means that we can embed \(X'\) and \(X'_n\) in the same space Z in such a way that

  1. (a)

    \(\rho =\rho _n\),

  2. (b)

    \(X'_n \subset (X')^{\varepsilon }\),

  3. (c)

    \(X' \subset (X'_n)^{\varepsilon }\),

  4. (d)

    for every \(A \subset X'_n\) that is measurable we have \(\mu _n(A) \le \mu (A^{\varepsilon })+\varepsilon \),

  5. (e)

    for every \(B \subset X'\) that is measurable we have \(\mu (B) \le \mu _n(B^{\varepsilon })+\varepsilon \).

This embedding provides a natural way to embed the measured metric spaces \(\overline{B_r}(X_n)\) and \(\overline{B_r}(X)\) in Z. We will deduce an upper bound for the GHP distance between these two hulls.

For all \(y \in \overline{B_r}(X_n)\) there is an \(x \in X'\) such that \(d(x,y) \le \varepsilon \). By Lemma 30 we have \(x \in \overline{B_{r+4\varepsilon }}(X)\), so by Lemma 28 and our choice of \(\varepsilon \), we have \(x \in \overline{B_r}(X)^{\delta }\). This proves \(\overline{B_r}(X_n) \subset \overline{B_r}(X)^{\delta +\varepsilon }\).

Similarly, let \(x \in \overline{B_r}(X)\). We have \(x \in \overline{B_{r-4\varepsilon }}(X)^{\delta }\) by Lemma 28 and our choice of \(\varepsilon \). Let \(z \in \overline{B_{r-4\varepsilon }}(X)\) be such that \(d(x,z) \le \delta \). There is a \(y \in X'_n\) such that \(d(y,z) \le \varepsilon \). By Lemma 30 we have \(y \in \overline{B_{r-4\varepsilon +4\varepsilon }}(X_n)\) and \(d(x,y) \le \delta +\varepsilon \), which proves \(\overline{B_r}(X) \subset \overline{B_r}(X_n)^{\delta +\varepsilon }\). Hence, in our embedding, the Hausdorff distance between \(\overline{B_r}(X)\) and \(\overline{B_r}(X_n)\) is less than \(\varepsilon +\delta \).

For all \(A \subset \overline{B_r}(X_n)\) measurable, we have \(\mu _n(A) \le \mu (A^{\varepsilon }) +\varepsilon =\mu \big ( A^{\varepsilon } \cap X' \big ) +\varepsilon \). By Lemma 30 we have the inclusion \(A^{\varepsilon } \cap X' \subset A^{\varepsilon } \cap \overline{B_{r+4\varepsilon }}(X)\), so we get

$$\begin{aligned} \mu _n(A)\le & {} \varepsilon +\mu \big ( A^{\varepsilon } \cap \overline{B_{r+4\varepsilon }}(X) \big )\\\le & {} \varepsilon + \mu \big ( A^{\varepsilon } \cap \overline{B_r}(X) \big ) + V(r+4\varepsilon )-V(r), \end{aligned}$$

where we recall that \(V(s)=\mu (\overline{B_s}(X))\) for all s.

Similarly, for all \(B \subset \overline{B_r}(X)\) measurable, we have

$$\begin{aligned} \mu (B)\le & {} \mu \big ( B \cap \overline{B_{r-4\varepsilon }}(X) \big ) + V(r) - V(r-4\varepsilon )\\\le & {} \varepsilon + \mu _n \big ( (B \cap \overline{B_{r-4\varepsilon }}(X))^{\varepsilon } \cap X'_n \big )+V(r) - V(r-4\varepsilon )\\\le & {} \varepsilon + \mu _n \big ( B^{\varepsilon } \cap \overline{B_{r-4\varepsilon }}(X)^{\varepsilon } \cap X'_n \big )+V(r) - V(r-4\varepsilon )\\\le & {} \varepsilon + \mu _n \big ( B^{\varepsilon } \cap \overline{B_r}(X_n) \big ) + V(r) - V(r-4\varepsilon ), \end{aligned}$$

where the last inequality uses Lemma 30.

Hence, our embedding of \(\overline{B_r}(X)\) and \(\overline{B_r}(X_n)\) gives the following bound for n large enough:

$$\begin{aligned} d_{GHP} \big ( \overline{B_r}(X), \overline{B_r}(X_n) \big ) \le \max \Big ( \varepsilon +\delta , \varepsilon +V(r+4\varepsilon )-V(r), \varepsilon + V(r) - V(r-4\varepsilon ) \Big ). \end{aligned}$$

By assumption (iii) in the statement of the proposition, the right-hand side can be made arbitrarily small, which proves the first point of Proposition 10. The second point is an obvious consequence of the first one.

Finally, in the case of compact, bipointed metric spaces \(\big ( (X_n, d_n), x_n,y_n, \mu _n \big )\) and \(\big ( (X, d), x,y, \mu \big )\) with \(r <d(x,y)\), the above proof still works. The first part of the proof (i.e. proving that the radii of the \(\big ( \overline{B_r}(X_n) \big )_{n \ge 0}\) are bounded) is not necessary anymore. We may apply the second part of the proof directly to X and \(X_n\) instead of the compact subsets \(X'\) and \(X'_n\). Note that if \(r>d(x,y)\), then \(\overline{B_r}(X)=X\) and \(\overline{B_r}(X_n)=X_n\) for n large enough, so the conclusion is immediate. \(\square \)

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Budzinski, T. The hyperbolic Brownian plane. Probab. Theory Relat. Fields 171, 503–541 (2018). https://doi.org/10.1007/s00440-017-0785-x

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Mathematics Subject Classification

  • 60F17
  • 60D05
  • 60C05