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Scaling limit for the ant in a simple high-dimensional labyrinth

Abstract

We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk in \(\mathbb {Z}^d\) converges to the Brownian motion on the integrated super-Brownian excursion when \(d>14\).

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Notes

  1. The lexicographical order assumes that the backbone is the leftmost branch of \(\mathcal {T}^{\text {GW}}_{\infty }\) Otherwise the vertices to the left of the backbone would never be listed.

  2. It might be that, for n small, some of the \(l_i^n,i=1,\dots ,K\) are not actually leaves, but in virtue of (7.7), for n sufficiently large, all of the \(l_i^n,n=1,\dots ,K\) are leaves with high probability.

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Acknowledgements

The authors would like to thank the referees whose work helped significantly improved the article.

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Correspondence to Gérard Ben Arous.

Appendix A: Basic estimates on branching random walks

Appendix A: Basic estimates on branching random walks

The goal for this section is to prove the existence of pivotal points for branching random walks (see Sect. 5.2 for notation).

Before starting, let us recall some estimates on simple random walks. We denote \(p_n(x,y)=P_x[X_n=y]\) the heat kernel of the simple random walk on \(\mathbb {Z}^d\). It verifies (see e.g. [50])

$$\begin{aligned} p_n(x,y) \le C n^{-d/2} \exp \left( -\frac{\left| x-y\right| ^2}{n}\right) . \end{aligned}$$
(A.1)

and (see e.g. [25])

$$\begin{aligned} c n^{-d/2} \exp \left( -\frac{\left| x-y\right| ^2}{n}\right) \le p_n(x,y)+ p_{n+1}(x,y) \end{aligned}$$
(A.2)

We also set \(G_{d}(x,y)=\mathbf{E}_x\Bigl [\sum _{n\in \mathbb {N}} {\mathbf 1}{\{X_n=y\}}\Bigr ]=\sum _{n\in \mathbb {N}} p_n(x,y)\) the Green function in \(\mathbb {Z}^d\). For \(d\ge 5\), we have (see e.g. Theorem 3.5 in [40]) for \(x\ne y\)

$$\begin{aligned} G_{d}(x,y)\le C \left| x-y\right| ^{-(d-2)}. \end{aligned}$$
(A.3)

1.1 A.1 Probability of intersection of critical branching random walks

We recall that \(\mathcal {B}(x)\) and \(\mathcal {T}^{\text {GW}}_*\) were defined in Sect. 5.2 . We have

Lemma A.1

Fix \(d\ge 5\). For any \(x,y\in \mathbb {Z}^d\) such that \(x\ne y\), we have

$$\begin{aligned} \mathbf{E}[ \left| \mathcal {B}(x)\cap \mathcal {B}(y)\right| ] \le C \left| x-y\right| ^{-(d-4)}. \end{aligned}$$

Proof

In order to estimate the number of intersections between \(\mathcal {B}(x)\) and \(\mathcal {B}(y)\), we shall do a union bound on all generations \(i^{(x)}\) of \(\mathcal {T}^{\text {GW}}_*(x)\) (resp. \(i^{(y)}\) of \(\mathcal {T}^{\text {GW}}_*(y)\)) and on all points \(z\in \mathcal {T}^{\text {GW}}_*(x)\) with \(\left| z\right| = i^{(x)}\) (resp. \(z\in \mathcal {T}^{\text {GW}}_*(y)\) with \(\left| z\right| =i^{(y)}\)). This yields, averaging over the randomness of the embedding,

$$\begin{aligned}&\mathbf{E}[ \left| \mathcal {B}(x)\cap \mathcal {B}(y)\right| ]\\&\quad \le \mathbf{E}\left[ \sum _{\begin{array}{c} i^{(x)} \in [0,H(\mathcal {T}^{\text {GW}}_*(x))] \\ i^{(y)}\in [0,H(\mathcal {T}^{\text {GW}}_*(y))] \end{array}} \sum _{\begin{array}{c} z\in \mathcal {T}^{\text {GW}}_*(x),\ \left| z\right| =i^{(x)} \\ z\in \mathcal {T}^{\text {GW}}_*(y),\ \left| z\right| =i^{(y)} \end{array}}\sum _{z\in \mathbb {Z}^d}p_{i^{(x)}}(x,z)p_{i^{(y)}}(y,z)\right] \\&\quad \le \mathbf{E}\left[ \sum _{\begin{array}{c} i^{(x)} \in [0,H(\mathcal {T}^{\text {GW}}_*(x))] \\ i^{(y)}\in [0,H(\mathcal {T}^{\text {GW}}_*(y))] \end{array}} \sum _{\begin{array}{c} z\in \mathcal {T}^{\text {GW}}_*(x),\ \left| z\right| =i^{(x)} \\ z\in \mathcal {T}^{\text {GW}}_*(y),\ \left| z\right| =i^{(y)} \end{array}}p_{i^{(x)}+i^{(y)}}(x,y)\right] \\&\quad \le \mathbf{E}\left[ \sum _{\begin{array}{c} i^{(x)} \in [0,H(\mathcal {T}^{\text {GW}}_*(x))] \\ i^{(y)}\in [0,H(\mathcal {T}^{\text {GW}}_*(y))] \end{array}} Z_{i^{(x)}}(\mathcal {T}^{\text {GW}}_*(x)) Z_{i^{(y)}}(\mathcal {T}^{\text {GW}}_*(y)) p_{i^{(x)}+i^{(y)}}(x,y)\right] , \end{aligned}$$

where \(Z_{n}(\mathcal {T}^{\text {GW}}_*(z))\) denotes the cardinality of \(\mathcal {T}^{\text {GW}}_*(z)\) at generation n.

Since the trees \(\mathcal {T}^{\text {GW}}_*(x)\) and \(\mathcal {T}^{\text {GW}}_*(y)\) are critical Galton–Watson trees with a special initial distribution \(\tilde{Z}_0\). The distribution of \(1+\tilde{Z}_0\) is a size-biased version of \(Z_1\) under \(\mathbf{P}\) and hence \(\mathbf{E}[\tilde{Z}_1]<\infty \) since \(\mathbf{E}[Z_1^2]<\infty \). Recalling that \(\mathbf{E}[Z_1]=1\), we know that \(\mathbf{E}[Z_{i^{(x)}}(\mathcal {T}^{\text {GW}}_*(x))]=\mathbf{E}[Z_{i^{(y)}}(\mathcal {T}^{\text {GW}}_*(y))]=\mathbf{E}[\tilde{Z}_1]<\infty \), which implies

$$\begin{aligned} \mathbf{E}[ \left| \mathcal {B}(x)\cap \mathcal {B}(y)\right| ]&\le C \sum _{i^{(x)}, i^{(y)}\in \mathbb {N}^2} p_{i^{(x)}+i^{(y)}}(x,y)\nonumber \\&\le C\sum _{k=0}^{\infty } k p_k(x,y) \nonumber \\&\le C \sum _{k=0}^{\infty } k k^{-d/2} \exp \left( -\frac{\left| x-y\right| ^2}{k}\right) , \end{aligned}$$
(A.4)

where we used (A.1). By using (A.2) as well as and (A.3) (valid for \(d\ge 5\)), we see that

$$\begin{aligned} \mathbf{E}[ \left| \mathcal {B}(x)\cap \mathcal {B}(y)\right| ] \le CG_{d-2}(x,y) \le C\left| x-y\right| ^{-(d-4)}. \end{aligned}$$

\(\square \)

Let us set

$$\begin{aligned} q(x,y)=\mathbf{P}[ \mathcal {B}(x)\cap \mathcal {B}(y)\ne \emptyset ]. \end{aligned}$$

By Markov’s inequality, we have the following.

Corollary A.2

Fix \(d\ge 5\). For any \(x\ne y\in \mathbb {Z}^d\), we have

$$\begin{aligned} q(x,y) \le C \left| x-y\right| ^{-(d-4)}. \end{aligned}$$

Let \(\mathcal {B}_{\ge n}^{(1)}(x)\) and \(\mathcal {B}^{(2)}(y)\) two independent sets. The first one is distributed as the image of the points at height larger that n of a modified Galton–Watson tree \(\mathcal {T}_*^{\text {GW}}(x)\) rooted at x (defined in Sect. 5.2) and \(\mathcal {B}^{(2)}(y)\) is simply distributed as \(\mathcal {B}(y)\).

Lemma A.3

For \(d\ge 5\) and any \(x,y\in \mathbb {Z}^d\) we have that

$$\begin{aligned} \mathbf{P}[\mathcal {B}^{(1)}_{\ge n}(x)\cap \mathcal {B}^{(2)}(y) \ne \emptyset ]\le C n^{-\frac{d-4}{2}}. \end{aligned}$$

Proof

We can use the same computations as in the previous proof to obtain a modified version of (A.4)

$$\begin{aligned} \mathbf{P}[\mathcal {B}^{(1)}_{\ge n}(x)\cap \mathcal {B}^{(2)}(y) \ne \emptyset ] \le C \sum _{k=n}^{\infty } k k^{-d/2} \exp \left( -\frac{\left| x-y\right| ^2}{k}\right) , \end{aligned}$$

and the right-hand side is maximized for \(x=y\). The proof follows. \(\square \)

Let us \(\{\mathcal {B}_{\ge n}(x)\cap \tilde{\mathcal {B}}(x) \ne \emptyset \}\) the event that there exists \(y\in \mathcal {T}_*^{\text {GW}}(x)\) such that

  1. 1.

    \(\left| y\right| \ge n\)

  2. 2.

    \(\phi _{\mathcal {T}_*^{\text {GW}}(x)}(y)\in \tilde{\mathcal {B}}(x)\) where \(\tilde{\mathcal {B}}(x)\) denotes the image after embedding of \(\mathcal {T}_*^{\text {GW}}(x)\setminus \overrightarrow{\mathcal {T}_*^{\text {GW}}(x)}_{z}\) with \(\overrightarrow{\mathcal {T}_*^{\text {GW}}(x)}_{z}\) being the descendant tree of the oldest ancestor of y which is not the root.

Lemma A.4

For \(d\ge 5\) and any \(x\in \mathbb {Z}^d\), we have

$$\begin{aligned} \mathbf{P}[\mathcal {B}_{\ge n}(x)\cap \tilde{\mathcal {B}}(x) \ne \emptyset ]\le C n^{-\frac{d-4}{2}}. \end{aligned}$$

Proof

Let us notice that \(\mathcal {B}_{\ge n}(x)\) and \(\tilde{\mathcal {B}}(x)\) are independent and that, by construction, the set \(\tilde{\mathcal {B}}(x)\) is obtained as the union of x and \(\tilde{Z}_0-1\) independent sets that we denote \(B^{i}\). Each one of these sets \(B^i\) has the law of \( \mathcal {B}^{(2)}(y)\) for some \(y\sim x \text { or }y=x\). Conditioning on the value of \(\tilde{Z}_0\), we see that

$$\begin{aligned} \mathbf{P}[\mathcal {B}_{\ge n}(x)\cap \tilde{\mathcal {B}}(x) \ne \emptyset ]&= 1- \mathbf{P}[\mathcal {B}_{\ge n}(x)\cap \tilde{\mathcal {B}}(x) = \emptyset ] \\&= 1- \mathbf{E}\Bigl [\prod _{i=2}^{\tilde{Z}_0} \mathbf{P}[\mathcal {B}_{\ge n}(x)\cap B^i = \emptyset ]\Bigr ]\\&\le 1- \mathbf{E}\Bigl [(\max _{y\sim x \text { or }y=x}\mathbf{P}[\mathcal {B}^{(1)}_{\ge n}(x)\cap \mathcal {B}^{(2)}(y)=\emptyset ])^{\tilde{Z}_0}\Bigr ] \\&\le 1-\Bigl (1-\mathbf{E}[\tilde{Z}_0] \max _{y\sim x \text { or }y=x}{} \mathbf{P}[\mathcal {B}^{(1)}_{\ge n}(x)\cap \mathcal {B}^{(2)}(y)\ne \emptyset ]\Bigr ) \\&\le C \max _{y\sim x \text { or }y=x}\mathbf{P}[\mathcal {B}^{(1)}_{\ge n}(x)\cap \mathcal {B}^{(2)}(y)\ne \emptyset ], \end{aligned}$$

where we used an expansion of the generating function of \(\tilde{Z}_0\) at 1 which is possible since \(\mathbf{E}[\tilde{Z}_0] <\infty \) because \(\mathbf{E}[Z^2]<\infty \).

The result then follows from Lemma A.3. \(\square \)

1.2 A.2 Probability of intersection of two IICBRW

We recall that \(\mathcal {B}(\alpha (R,\infty ))\) for \(R>0\) was defined in Sect. 5.2. Let us first prove the following lemma.

Lemma A.5

Fix \(d\ge 7\). For \(\alpha (n)\) a simple random walk started at 0, we have for any \(R>0\)

$$\begin{aligned} \mathbf{P}[\mathcal {B}(0)\cap \mathcal {B}(\alpha ((R,\infty ))) \ne \emptyset ] \le C R^{-(d-6)/2}. \end{aligned}$$

Proof

We have by (A.1) and Corollary A.2

$$\begin{aligned} \mathbf{P}[\mathcal {B}(0)\cap \mathcal {B}(\alpha ((R,\infty ))) \ne \emptyset ]&\le \sum _{n>R} \sum _{y\in \mathbb {Z}^d} p_n(0,y) q(0,y) \\&\le C \sum _{n>R} \sum _{y\in \mathbb {Z}^d} n^{-d/2} \exp \left( -\frac{\left| y\right| ^2}{n}\right) \left| y\right| ^{-(d-4)} \\&\le C \sum _{n>R} n^{-d/2} \sum _{k=0}^{\infty } k^{d-1} k^{-(d-4)} \exp \left( -\frac{\left| k\right| ^2}{n}\right) , \end{aligned}$$

where, to obtain the last line, we have partitioned \(\mathbb {Z}^d\) into spheres. From here, we may notice that

$$\begin{aligned} \sum _{k=0}^{\infty } k^3 \exp \left( -\frac{\left| k\right| ^2}{n}\right) \le C \int _0^{\infty } x^3\exp \left( -\frac{x^2}{n}\right) dx \le C n^{2}\int _0^{\infty } u^3++-u^2)du, \end{aligned}$$

which leads us to the following bound

$$\begin{aligned} \mathbf{P}[\mathcal {B}(0)\cap \mathcal {B}(X_{(R,\infty )}) \ne \emptyset ]&\le C \sum _{n>R} n^{-d/2} n^{2} \\&\le C\sum _{n>R} n^{-(d-4)/2} \\&\le CR^{-(d-6)/2}. \end{aligned}$$

\(\square \)

Our main estimate for this section is the following

Lemma A.6

Fix \(d\ge 9\). For \((\alpha (n))_{n\in \mathbb {Z}}\) a simple random walk started at 0, we have,

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((-\infty , 0)))\cap \mathcal {B}(\alpha ((R,\infty ))) \ne \emptyset ] \le C R^{-(d-8)/2}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((-\infty , 0)))\cap \mathcal {B}(\alpha ((R,\infty ))) \ne \emptyset ]&\le \sum _{l=0}^{\infty } \mathbf{P}[\mathcal {B}(\alpha (-l))\cap \mathcal {B}(\alpha ((R,\infty ))) \ne \emptyset ] \\&\le \sum _{l=0}^{\infty } \mathbf{P}[\mathcal {B}(0)\cap \mathcal {B}(\alpha ((R+l,\infty ))) \ne \emptyset ], \end{aligned}$$

where we use translation invariance. Then, we can see by Lemma A.5

$$\begin{aligned} \sum _{l=0}^{\infty } \mathbf{P}[\mathcal {B}(0)\cap \mathcal {B}(\alpha ((R+l,\infty ))) \ne \emptyset ]\le & {} C\sum _{l=0}^{\infty } (R+l)^{-(d-6)/2} \\\le & {} C \sum _{j=R}^{\infty } j^{-(d-8)/2}\\\le & {} C R^{-(d-8)/2}, \end{aligned}$$

and the lemma follows. \(\square \)

1.3 A.3 Existence of pivotal points

The notion of pivotal point was defined in Sect. 5.2. The goal of this part is to prove the existence of such points in high enough dimensions. Let us write \([x]:=1\wedge \left| \left| x\right| \right| \).

We can prove the following.

Lemma A.7

Fix \(d>6\). For \((\alpha (n))_{n\in \mathbb {Z}}\) a simple random walk started at 0, we have for any \(x\in \mathbb {Z}^d \)

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \mathcal {B}(x) \ne \emptyset ] \le C [x]^{-(d-6)}. \end{aligned}$$

Proof

We have that

$$\begin{aligned}&\mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \mathcal {B}(x) \ne \emptyset ] \\&\quad \le \sum _{n\ge 0} \sum _{y\in \mathbb {Z}^d} p_n(0,y)q(x,y) \\&\quad \le \sum _{y\in \mathbb {Z}^d} G_d(0,y) q(x,y) \\&\quad \le \sum _{y\in \mathbb {Z}^d} [y]^{-(d-2)}[x-y]^{-(d-4)}, \end{aligned}$$

where we used (A.3) and Corollary A.2. We can then use Proposition 1.7(i) of [27] to see that for \(d>6\)

$$\begin{aligned} \sum _{y\in \mathbb {Z}^d} [y]^{-(d-2)}[x-y]^{-(d-4)}\le C [x]^{-(d-6)}. \end{aligned}$$

\(\square \)

For \(x\in \mathbb {Z}^d\) and \(A\subset \mathbb {Z}^d\), we used the notation \(x+A\) to denote \(\{y\in \mathbb {Z}^d, y=x+z \text { for } z\in A\}\). Let us prove that

Lemma A.8

Fix \(d>8\). For \((\alpha (n))_{n\in \mathbb {Z}}\) a simple random walk started at 0, we have for any x

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \{x+ \mathcal {B}(\alpha ((-\infty ,0))) \} \ne \emptyset ] \le C [x]^{-(d-8)}. \end{aligned}$$

Proof

We have by Lemma A.7

$$\begin{aligned}&\mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \{x+ \mathcal {B}(\alpha ((-\infty ,0))) \} \ne \emptyset ] \\&\quad \le \sum _{i=1}^{\infty }{} \mathbf{P}[\mathcal {B}(\alpha (i))\cap \{x+ \mathcal {B}(\alpha ((-\infty ,0))) \} \ne \emptyset ] \\&\quad \le \sum _{i=1}^{\infty } \sum _{y\in \mathbb {Z}^d} p_i(0,y) \mathbf{P}[\mathcal {B}(y)\cap \{x+ \mathcal {B}(\alpha ((-\infty ,0))) \} \ne \emptyset ] \\&\quad \le C \sum _{i=1}^{\infty } \sum _{y\in \mathbb {Z}^d} p_i(0,y) [x-y]^{-(d-6)} \\&\quad \le C \sum _{y\in \mathbb {Z}^d} G_d(0,y) [x-y]^{-(d-6)} \\&\quad \le C \sum _{y\in \mathbb {Z}^d} [y]^{-(d-2)} [x-y]^{-(d-6)}, \end{aligned}$$

where we used (A.3) in the last line. We can then use Proposition 1.7(i) of [27] to see that for \(d>8\) we have

$$\begin{aligned} \sum _{y\in \mathbb {Z}^d} [y]^{-(d-2)} [x-y]^{-(d-6)} \le C [x]^{-(d-8)}. \end{aligned}$$

which implies the lemma. \(\square \)

Let us prove Lemma 5.9.

Proof of Lemma 5.9

Let us describe an event on which 0 is a pivotal point and then prove that this event has positive probability. Let us denote \(E_k\) the event that

  1. 1.

    \(\alpha (i)-\alpha (i-1)=e_1\) for \(i\in [-k+1,k]\) and \(\mathcal {B}(\alpha ([-k,k])) = \{i e_1, \ \text {for } i\in [-k,k]\}\).

  2. 2.

    \(\mathcal {B}(\alpha ((k,\infty )))\cap \mathcal {B}(\alpha ((-\infty ,-k))) = \emptyset \).

  3. 3.

    \(\mathcal {B}(\alpha ((k,\infty ))) \cap \{i e_1, \left| i\right| \le k/2 \}=\emptyset \).

  4. 4.

    \(\mathcal {B}(\alpha ((-\infty ,-k))) \cap \{i e_1, \left| i\right| \le k/2 \}=\emptyset \).

It is clear that \(E_k \subset \{0\text { is a pivotal point}\}\). We can also notice the sub-event in the first line has a positive probability \(c_k\) for any k and is independent of the events on the other lines. Conditioning on this event we can see that

$$\begin{aligned} \mathbf{P}[0\text { is a pivotal point}] \ge \mathbf{P}[E_k] \ge c_k \mathbf{P}[F_k], \end{aligned}$$
(A.5)

where we used that the law of \(\mathcal {B}(\alpha ((k,\infty ))) -\alpha (k)\) and \(\mathcal {B}(\alpha ((0,\infty )))\) are the same and where \(F_k\) is the event that

  1. 1.

    \(\mathcal {B}(\alpha ((0,\infty )))\cap \{2k e_1+ \mathcal {B}(\alpha ((-\infty ,0)))\}=\emptyset \).

  2. 2.

    \(\mathcal {B}(\alpha ((0,\infty ))) \cap \{i e_1, \text { for } i\in [-k/2, -3k/2]\}=\emptyset \).

  3. 3.

    \(\mathcal {B}(\alpha ((-\infty ,0))) \cap \{i e_1, \text { for } i\in [k/2, 3k/2]\}=\emptyset \).

Using Lemma A.7 for \(x =ke_1,(k+1)e_1, \ldots \) and the fact that \(d> 8\) we see that for any k

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \{k' e_1,\ \text { for } k'\ge k\} \ne \emptyset ] \le C k^{-(d-8)}. \end{aligned}$$

this means we can choose \(k_0\) such that for \(k\ge k_0\)

$$\begin{aligned} \mathbf{P}[\mathcal {B}((0,\infty ))\cap \{j e_1,\ j\ge k/2 \} =\emptyset ] >\frac{5}{6}, \end{aligned}$$

and similarly

$$\begin{aligned} \mathbf{P}[\mathcal {B}((-\infty ,0))\cap \{j e_1,\ j\le -k/2 \} =\emptyset ] >\frac{5}{6}, \end{aligned}$$

which by independence yields

$$\begin{aligned} \mathbf{P} [\mathcal {B}((0,\infty ))\cap \{j e_1,\ j\ge k/2 \} =\emptyset ,\ \mathcal {B}((-\infty ,0))\cap \{j e_1,\ j\le -k/2 \} =\emptyset ] >\frac{25}{36}. \end{aligned}$$

Now we can use Lemma A.8 to see that there exists \(k_1\) such that for \(k\ge k_1\)

$$\begin{aligned} \mathbf{P}[\mathcal {B}(\alpha ((0,\infty )))\cap \{2k e_1+ \mathcal {B}(\alpha ((-\infty ,0))) \} = \emptyset ] \ge \frac{2}{3}. \end{aligned}$$

Hence for \(k_2= \max (k_1,k_0)\), we have

$$\begin{aligned} \mathbf{P}[F_{k_2}] \ge \frac{1}{6}, \end{aligned}$$

and thus by (A.5) means that \(\mathbf{P}[0 \text { is pivotal}]>0\). \(\square \)

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Ben Arous, G., Cabezas, M. & Fribergh, A. Scaling limit for the ant in a simple high-dimensional labyrinth. Probab. Theory Relat. Fields 174, 553–646 (2019). https://doi.org/10.1007/s00440-018-0876-3

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Keywords

  • Random walk
  • Random environments
  • Branching random walk
  • Super-process
  • Spatial tree

Mathematics Subject Classification

  • Primary 60K37
  • Secondary 82D30