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Geometry of the minimal spanning tree of a random 3-regular graph

Abstract

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph (Addario-Berry et al. in Ann Probab 45(5):3075–3144, 2017). In this work, we show that the MST constructed by assigning i.i.d. continuous edge weights to either the random (simple) 3-regular graph or the 3-regular configuration model on n vertices, endowed with the tree distance scaled by \(n^{-1/3}\) and the uniform probability measure on the vertices, converges in distribution with respect to Gromov–Hausdorff–Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in Addario-Berry et al. (2017) up to a scaling factor of \(6^{1/3}\). Our proof relies on a novel argument that proceeds via a comparison between a 3-regular configuration model and the largest component in the critical Erdős–Rényi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

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Notes

  1. Here, the superscript ‘\({{\,\mathrm{er}\,}}\)’ is being used to refer to the Erdős–Rényi random graph. The reason behind using this notation will become clear in Sect. 4.3.

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Acknowledgements

The authors thank two anonymous referees whose careful reading and detailed comments led to significant improvements in the paper. LAB was supported in part by an NSERC Discovery Grant and Discovery Accelerator Supplement, and by an FRQNT Team Grant, during the preparation of this research. SS was partially supported by the Infosys Foundation, Bangalore and by MATRICS grant MTR/2019/000745 from SERB.

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Appendix A

Appendix A

Our aim in this section is to briefly describe the ideas needed to prove Theorems 3.11 and 3.13 .

1.1 Sketch of proof of Theorem 3.11

Suppose \({\mathbf {t}}\) is a rooted tree with vertices labeled by [m] and let \(R(\cdot ,{\mathbf {t}})\) be as in (6.59). For \(s\ge 1\), let

$$\begin{aligned} A_s({\mathbf {t}}):=\big \{\big (v_1, u_1,\ldots , v_s,u_s\big ) \ :&\ 1\le v_1\le \ldots \le v_s\le m,\ u_i\in R(v_i,{\mathbf {t}}), \nonumber \\&\ \text { if } i< j \text { and }v_i=v_j\text { then }u_i<u_j\big \}. \end{aligned}$$

Note that \(s!\times |A_s({\mathbf {t}})|\le |A_1({\mathbf {t}})|^s\). Let \(T_m\) denote a uniform rooted labeled tree on [m], and let \({\overline{T}}_m\) be distributed as

$$\begin{aligned} {\mathbb {P}}\big ({\overline{T}}_m={\mathbf {t}}\big )=\frac{{{\,\mathrm{{\mathbb {P}}}\,}}\big (T_m={\mathbf {t}}\big )\cdot |A_s({\mathbf {t}})|}{{\mathbb {E}}\big (|A_s(T_m)|\big )}. \end{aligned}$$

Then we have the following decomposition of \({\mathscr {H}}_{m, s}\):

Theorem A.1

Fix \(s\ge 1\). Sample \({\overline{T}}_m\), and conditional on the realization, sample \(\big ({\overline{v}}_{1,m}, {\overline{u}}_{1,m},\ldots , {\overline{v}}_{s,m},{\overline{u}}_{s,m}\big )\) from \(A_s({\overline{T}}_m)\) uniformly. Place an edge between \({\overline{v}}_{i,m}\) and \({\overline{u}}_{i,m}\) for \(1\le i\le s\), and then forget about the root of \({\overline{T}}_m\). Call the resulting graph \(\overline{{\mathscr {H}}}_{m,s}\). Then \(\overline{{\mathscr {H}}}_{m,s}{\mathop {=}\limits ^{\text {d}}}{\mathscr {H}}_{m, s}\).

This can be seen as follows: Consider a simple, connected, rooted graph G on [m] with \(\text {sp}(G)=s\). Let \({\mathbf {t}}\) be the tree constructed by following a depth-first exploration of G starting at its root, and let \(v_i, u_i\), \(1\le i\le s\), be the endpoints of the s edges that need to be added to \({\mathbf {t}}\) to recover G. We can arrange \(v_1, u_1,\ldots , v_s, u_s\) in a unique way so that the resulting sequence becomes an element of \(A_s({\mathbf {t}})\). It thus follows that the set of simple, connected, rooted graphs on [m] having s surplus edges is in bijective correspondence with the set

$$\begin{aligned} \big \{ ({\mathbf {t}}, v_1, u_1,\ldots , v_s, u_s)\ :\ {\mathbf {t}}\text { rooted tree on }[m], (v_1, u_1,\ldots , v_s, u_s)\in A_s({\mathbf {t}}) \big \} . \end{aligned}$$
(A.1)

Then one can show that if we root \({\mathscr {H}}_{m, s}\) at a uniform vertex, then its corresponding element in the set (A.1) will be distributed as \(\big ({\overline{T}}_m, {\overline{v}}_{1,m},{\overline{u}}_{1,m},\ldots ,{\overline{v}}_{s,m},{\overline{u}}_{s,m}\big )\). We omit the details as similar ideas have already been used in [3, 26, 27].

For any tree \({\mathbf {t}}\) on [m] rooted at \(\rho \), endow the children of each vertex in \({\mathbf {t}}\) with the linear order induced by their labels. Let \(\rho =w_0, w_1,\ldots , w_{m-1}\) be the vertices of \({\mathbf {t}}\) in order of appearance in a depth-first exploration of \({\mathbf {t}}\) using the above order. Let \({{\,\mathrm{Ht}\,}}_{{\mathbf {t}}}:[0,m]\rightarrow {\mathbb {R}}\) be the height function of \({\mathbf {t}}\) given by \({{\,\mathrm{Ht}\,}}_{{\mathbf {t}}}(m)=0\), and

$$\begin{aligned} {{\,\mathrm{Ht}\,}}_{{\mathbf {t}}}(x)={{\,\mathrm{ht}\,}}(w_{\lfloor x\rfloor },{\mathbf {t}}),\ \ \ x\in [0,m). \end{aligned}$$

The following lemma is a collection of some standard results about \(T_m\):

Lemma A.2

(i) :

The following convergences hold:

$$\begin{aligned}&m^{-1/2}{{\,\mathrm{Ht}\,}}_{T_m}\big (m\cdot \big ){\mathop {\longrightarrow }\limits ^{\text {d}}}2{\mathbf {e}}(\cdot ), \ \ \ \ \text { and} \nonumber \\&m^{-1/2}\max _{v\in [m]}\big |2\big |R(v, T_m)\big |-{{\,\mathrm{ht}\,}}(v, T_m)\big |{\mathop {\longrightarrow }\limits ^{\text {P}}}0, \end{aligned}$$
(A.2)

where the convergence in (A.2) is w.r.t. the Skorohod \(J_1\) topology.

(ii) :

For all \(m\ge 1\), \({{\,\mathrm{{\mathbb {P}}}\,}}\big ({{\,\mathrm{ht}\,}}(T_m)\ge x\sqrt{m}\big )\le c x^3\exp \big (-x^2/2\big )\).

(iii) :

For all \(x\ge 0\) and \(m\ge 1\),

$$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big (\max _{v\in [m]}|R(v, T_m)|\ge x\sqrt{m}\big )\le c\exp (-c'x^2). \end{aligned}$$

Using the bounds \(|A_s(T_m)|\times s!\le |A_1(T_m)|^s\) and \(|A_1(T_m)|\le m\cdot \max _{v\in [m]}|R(v, T_m)|\), we further have

$$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big (|A_s(T_m)|\ge x m^{3s/2}\big )\le c\exp \big (-c'x^{2/s}\big ) \end{aligned}$$

for any \(s\ge 1\), \(x\ge 0\), and \(m\ge 1\).

(iv) :

For any \(s\ge 1\), \(m^{-3s/2}\big (|A_1(T_m)|^s-|A_s(T_m)|\times s!\big ){\mathop {\longrightarrow }\limits ^{\text {P}}}0\).

Lemma A.2((i)) follows from [69]. ((ii)) follows from [66, Corollary 1]. ((iii)) is the content of [3, Lemma 13]. The proof of ((iv)) is similar to that of [26, Lemma 7.3 (iii)].

Sketch of proof of (3.9)

In view of Theorem A.1, \(s\cdot \big (\text {ht}({\overline{T}}_m)+1\big )\) dominates \(L({\mathscr {H}}_{m, s})\) stochastically for any \(s\ge 1\). Thus, (3.9) follows from Lemma A.2 ((ii)) and ((iii)). \(\square \)

To prove the other assertions in Theorem 3.11 it will be convenient to work with two slightly different spaces \({\mathscr {H}}_{m, s}^{\circ }\) and \({\mathscr {H}}_{m, s}^{\dagger }\) which we define next. Recall the notation \(R(\cdot ,\cdot ,\cdot )\) from (6.58). Sample \(T_m^{\circ }\) according to distribution

$$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big (T_m^{\circ }={\mathbf {t}}\big )=\frac{{{\,\mathrm{{\mathbb {P}}}\,}}(T_m={\mathbf {t}})\cdot |A_1({\mathbf {t}})|^s}{{\mathbb {E}}\big [|A_1(T_m)|^s\big ]}~,\ \ \ \ {\mathbf {t}}\text { rooted tree on }[m]. \end{aligned}$$
(A.3)

Conditional on \(T_m^{\circ }\), sample an i.i.d. sequence of triples \((v_{i,m}^{\circ }, u_{i,m}^{\circ }, f_{i,m}^{\circ })\), \(1\le i\le s\), where

$$\begin{aligned}&{{\,\mathrm{{\mathbb {P}}}\,}}\big (v_{i,m}^{\circ }=v\mid T_m^{\circ }\big )=|R(v, T_m^{\circ })|\big /|A_1(T_m^{\circ })|,\ \ \ v\in [m],\\&{{\,\mathrm{{\mathbb {P}}}\,}}\big (u_{i,m}^{\circ }=u\mid T_m^{\circ },v_{i,m}^{\circ }\big ) =|R(u, v_{i,m}^{\circ }, T_m^{\circ })|\big /|R(v, T_m^{\circ })|,\ \ \ u\in \big \{{\mathop {(v_{i,m}^{\circ }) }\limits ^{\longleftarrow {\scriptscriptstyle (k)}}}\ :\ 1\le k\le {{\,\mathrm{ht}\,}}(v_{i,m}^{\circ })\big \},\ \ \text { and}\\&{{\,\mathrm{{\mathbb {P}}}\,}}\big (f_{i,m}^{\circ }=f\mid T_m^{\circ },v_{i,m}^{\circ }, u_{i,m}^{\circ }\big ) =1\big /|R(u_{i,m}^{\circ }, v_{i,m}^{\circ }, T_m^{\circ })|,\ \ \ f\in R(u_{i,m}^{\circ }, v_{i,m}^{\circ }, T_m^{\circ }). \end{aligned}$$

Let \({\mathscr {H}}_{m, s}^{\dagger }\) (resp. \({\mathscr {H}}_{m, s}^{\circ }\)) be the space obtained by adding an edge between \(v_{i,m}^{\circ }\) and \(f_{i,m}^{\circ }\) (resp. between \(v_{i,m}^{\circ }\) and \(u_{i,m}^{\circ }\)) for \(1\le i\le s\), and then forgetting about the root of \(T_m^{\circ }\). It follows from Lemma A.2 ((iii)) and ((iv)) that the total variation distance between the laws of \({\overline{{\mathscr {H}}}}_{m, s}\) (as defined in Theorem A.1) and \({\mathscr {H}}_{m, s}^{\dagger }\) tends to zero as \(m\rightarrow \infty \). It thus follows from Theorem A.1 that there exists a coupling of \({\mathscr {H}}_{m, s}\) and \({\mathscr {H}}_{m, s}^\dagger \) such that

$$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big ({\mathscr {H}}_{m, s}\ne {\mathscr {H}}_{m, s}^\dagger \big )\rightarrow 0,\ \ \text { as }\ \ m\rightarrow \infty . \end{aligned}$$
(A.4)

We will now recall an alternate construction of \({\mathscr {H}}^{(s)}\) which is essentially given in [3]; see also the discussion below [2, Equation (1)]. We first introduce some notation. For any \(f:[0,1]\rightarrow {\mathbb {R}}, x\in [0,1]\), and \(h>0\), let

$$\begin{aligned} {{\,\mathrm{prev}\,}}(x,h;f)=\sup \big \{y\in [0,x) : f(y)=h\big \}, \ \ \ \text { and } \ \ \ {{\,\mathrm{next}\,}}(x,h;f)=\inf \big \{y\in (x,1] : f(y)<h\big \}, \end{aligned}$$

where \(\sup \{\ \}=-\infty \) and \(\inf \{\ \}=\infty \) by convention.

Construction A.3

(Alternate construction of \({\mathscr {H}}^{(s)}\)) Fix an integer \(s\ge 2\).

  1. (a)

    Sample \({\mathbf {e}}^{\circ }\) with law given by

    $$\begin{aligned} {\mathbb {E}}\big [f({\mathbf {e}}^{\circ })\big ]=\frac{{\mathbb {E}}\big [f({\mathbf {e}})\big (\int _0^1 {\mathbf {e}}(t)dt\big )^s\big ]}{{\mathbb {E}}\big [\big (\int _0^1 {\mathbf {e}}(t)dt\big )^s\big ]}. \end{aligned}$$
  2. (b)

    Conditional on \({\mathbf {e}}^{\circ }\), sample i.i.d. points \(y_1^{\circ },\ldots , y_s^{\circ }\) having density \({\mathbf {e}}^{\circ }(y)\big /\int _0^1 {\mathbf {e}}^{\circ }(t)dt \).

  3. (c)

    Conditional on the above, sample \(h_1^{\circ },\ldots ,h_s^{\circ }\) independently, where \(h_i^{\circ }\sim \text {Unif}[0,{\mathbf {e}}^{\circ }(y_i^\circ )]\). Set \(x_i^\circ ={{\,\mathrm{prev}\,}}(y_i^\circ , h_i^\circ ; {\mathbf {e}}^\circ )\).

  4. (d)

    Form the quotient space \({\mathscr {T}}_{{\mathbf {e}}^\circ }/\sim \), where \(\sim \) is the equivalence relation under which \(q_{{\mathbf {e}}^\circ }(x_i^\circ )\sim q_{{\mathbf {e}}^\circ }(y_i^\circ )\), \(1\le i\le s\).

Then \({\mathscr {H}}^{(s)}{\mathop {=}\limits ^{\text {d}}}2\cdot \big ({\mathscr {T}}_{{\mathbf {e}}^\circ }/\sim \big )\).

Now observe that \({\mathscr {H}}_{m, s}^{\circ }\) has a similar alternate construction: First sample \(T_m^\circ \) as in (A.3). Let \(w_0,\ldots ,w_{m-1}\) be the vertices of \(T_m^\circ \) in order of appearance in a depth-first exploration of \(T_m^\circ \). Let \({{\,\mathrm{Ht}\,}}^\circ \) be the height function of \(T_m^\circ \). Conditional on \(T_m^\circ \), sample i.i.d. random variables \(y_{1,m}^\circ ,\ldots ,y_{s,m}^\circ \), where

$$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big (y_{i,m}^\circ =j\mid T_m^\circ \big )=|R(w_j, T_m^\circ )|\big /|A_1(T_m^\circ )|,\ \ \ 1\le j\le m-1. \end{aligned}$$

Conditional on the above, sample \(h_{1,m}^\circ ,\ldots , h_{s,m}^\circ \) independently via

$$\begin{aligned}&{{\,\mathrm{{\mathbb {P}}}\,}}\big (h_{i,m}^\circ ={{\,\mathrm{Ht}\,}}^\circ (y_{i,m}^\circ )-k\ \big |\ T_m^\circ ,y_{1,m}^\circ ,\ldots ,y_{s,m}^\circ \big ) \\&\quad = \frac{|R\big ({\mathop {v\ \ \ }\limits ^{\leftarrow \scriptscriptstyle {(k)}}}, v, T_m^\circ \big )|}{|R(v, T_m^\circ )|},\ \ \ 1\le k\le {{\,\mathrm{Ht}\,}}^\circ (y_{i,m}^\circ ), \end{aligned}$$

where \(v=w_{y_{i,m}^\circ }\). Let \(x_{i,m}^\circ ={{\,\mathrm{prev}\,}}(y_{i,m}^\circ , h_{i,m}^{\circ }; {{\,\mathrm{Ht}\,}}^\circ )-1\). Then \({\mathscr {H}}_{m, s}^\circ \) has the same distribution as the space obtained by placing an edge in \(T_m^\circ \) between \(w_{y_{i,m}^\circ }\) and \(w_{x_{i,m}^\circ }\) for \(1\le i\le s\).

Sketch of proof of (3.8)

Using Lemma A.2 ((i)) and ((iii)), it can be shown that the following convergences hold jointly:

$$\begin{aligned} \frac{1}{\sqrt{m}}{{\,\mathrm{Ht}\,}}^\circ \big (m\cdot \big ){\mathop {\longrightarrow }\limits ^{\text {d}}}2{\mathbf {e}}^\circ (\cdot ), \ \ \text { and }\ \ \Big (\frac{x_{i,m}^\circ }{m}, \frac{y_{i,m}^\circ }{m}, \frac{h_{i,m}^\circ }{\sqrt{m}}\Big ) {\mathop {\longrightarrow }\limits ^{\text {d}}}\big (x_i^\circ , y_i^\circ , 2h_i^\circ \big ),\ \ 1\le i\le s \end{aligned}$$
(A.5)

as \(m\rightarrow \infty \). Using Construction A.3 and the above alternate construction of \({\mathscr {H}}_{m, s}^\circ \), it is now routine to prove the assertion in (3.8) for \({\mathscr {H}}_{m, s}^\circ \), from which it follows that the same is true for \({\mathscr {H}}_{m, s}^\dagger \). The desired result now follows from (A.4). \(\square \)

Let \(y_{(i),m}^\circ \) (resp. \(y_{(i)}^\circ \)), \(1\le i\le s\), be \(y_{i,m}^\circ \) (resp. \(y_i^\circ \)), \(1\le i\le s\), arranged in an increasing order. For \(1\le i\le s-1\) define \(z_{i,m}^\circ \) and \(z_i^\circ \) via

$$\begin{aligned}&z_{i,m}^\circ =\min \Big \{t\in [y_{(i),m}^\circ , y_{(i+1),m}^\circ ]\ :\ {{\,\mathrm{Ht}\,}}^\circ (t)=\min \big \{ {{\,\mathrm{Ht}\,}}^\circ (a) : y_{(i),m}^\circ \le a\le y_{(i+1),m}^\circ \big \} \Big \},\ \ \text { and}\\&{\mathbf {e}}^\circ (z_i^\circ )=\inf \big \{ {\mathbf {e}}^\circ (t) : y_{(i)}^\circ \le t\le y_{(i+1)}^\circ \big \}. \end{aligned}$$

Further, define

$$\begin{aligned}&x_{i,m}^{\circ , +}={{\,\mathrm{next}\,}}\big (x_{i,m}^\circ , h_{i,m}^\circ +1;\ {{\,\mathrm{Ht}\,}}^\circ \big ),\ \ x_i^{\circ , +}={{\,\mathrm{next}\,}}\big (x_i^\circ , h_i^\circ ; {\mathbf {e}}^\circ \big ),\ \ 1\le i\le s,\\&z_{i,m}^{\circ , -}={{\,\mathrm{prev}\,}}\big (z_{i,m}^\circ , {{\,\mathrm{Ht}\,}}^\circ (z_{i,m}^\circ )-1;\ {{\,\mathrm{Ht}\,}}^\circ \big )-1,\ \ z_i^{\circ , -}={{\,\mathrm{prev}\,}}\big (z_i^\circ , {\mathbf {e}}^\circ (z_i^\circ ); {\mathbf {e}}^\circ \big ),\ \ 1\le i\le s-1,\\&z_{i,m}^{\circ , +}={{\,\mathrm{next}\,}}\big (z_{i,m}^\circ , {{\,\mathrm{Ht}\,}}^\circ (z_{i,m}^\circ );\ {{\,\mathrm{Ht}\,}}^\circ \big )-1,\ \ z_i^{\circ , +}={{\,\mathrm{next}\,}}\big (z_i^\circ , {\mathbf {e}}^\circ (z_i^\circ ); {\mathbf {e}}^\circ \big ),\ \ 1\le i\le s-1. \end{aligned}$$

Sketch of proof of (3.11)

From (A.5) it follows that the following convergence holds jointly with the convergence in (A.5): As \(m\rightarrow \infty \),

$$\begin{aligned} \frac{x_{i,m}^{\circ , +}}{m}{\mathop {\longrightarrow }\limits ^{\text {d}}}x_i^{\circ , +},\ 1\le i\le s,\ \text { and }\ \frac{1}{m}\big (z_{i,m}^\circ ,\ z_{i,m}^{\circ , -},\ z_{i,m}^{\circ , +}\big ){\mathop {\longrightarrow }\limits ^{\text {d}}}\big (z_i^\circ , z_i^{\circ , -}, z_i^{\circ , +}\big ),\ 1\le i\le s-1. \end{aligned}$$
(A.6)

Arrange \(x_i^\circ , x_i^{\circ , +}, y_i^\circ \), \(1\le i\le s\), and \(z_i^\circ , z_i^{\circ , -}, z_i^{\circ , +}\), \(1\le i\le s-1\), (resp. \(x_{i,m}^\circ , x_{i,m}^{\circ , +}, y_{i,m}^\circ \), \(1\le i\le s\), and \(z_{i,m}^\circ , z_{i,m}^{\circ , -}, z_{i,m}^{\circ , +}\), \(1\le i\le s-1\)) in increasing order as \(a_1,\ldots ,a_{6s-3}\) (resp. as \(a_{1,m},\ldots ,a_{6s-3, m}\)). Let

$$\begin{aligned} \Delta _j=a_{j+1}-a_j,\ \ \text { and }\ \ \Delta _{j,m}=a_{j+1,m}-a_{j,m},\ \ 1\le j\le 6s-4. \end{aligned}$$

Then it follows from (A.6) and the second convergence in (A.5) that

$$\begin{aligned} \big (\Delta _{j,m},\ 1\le j\le 6s-4\big ) {\mathop {\longrightarrow }\limits ^{\text {d}}}\big (\Delta _j,\ 1\le j\le 6s-4\big ),\ \ \text { as }\ \ m\rightarrow \infty \end{aligned}$$
(A.7)

jointly with (A.5) and (A.6).

Recall the notation used in (3.11), and note that there exists a partition \({\mathscr {P}}=\{{\mathscr {P}}_1,\ldots ,{\mathscr {P}}_r\}\) of \([6s-4]\) that depends only on the realizations of \({\mathbf {e}}^\circ \) and \(x_i^\circ ,y_i^\circ \), \(1\le i\le s\), such that

$$\begin{aligned} \big (\mu ^{(s)}\big ({\mathscr {T}}_i'\big ), 1\le i\le r\big ) {\mathop {=}\limits ^{\text {d}}}\big (\sum _{j\in {\mathscr {P}}_i}\Delta _j,\ 1\le i\le r\big ). \end{aligned}$$
(A.8)

Further, it follows from (A.5) that for large m, the vector consisting of the numbers of vertices in \({\mathscr {H}}_{m, s}^\circ \) that are connected to the different elements of \(e({\mathscr {H}}_{m, s}^\circ )\) is given by \(\big (\sum _{j\in {\mathscr {P}}_i}\Delta _{j,m},\ 1\le i\le r\big )\), where the common endpoints of multiple \(e\in e({\mathscr {H}}_{m, s}^\circ )\) and the vertices in their pendant subtrees have been accounted for in \(\sum _{j\in {\mathscr {P}}_i}\Delta _{j,m}\) for exactly one value of i in a specific way. Using (A.7) and (A.8), we get the analogue of (3.11) for \({\mathscr {H}}_{m, s}^\circ \) for the above specific way of assigning the common endpoints of multiple \(e\in e({\mathscr {H}}_{m, s}^\circ )\) and the vertices in their pendant subtrees to the different terms \(\sum _{j\in {\mathscr {P}}_i}\Delta _{j,m}\).

This together with (A.4) would complete the proof if we can show that the sizes of the pendant subtrees of the common endpoints of multiple \(e\in e({\mathscr {H}}_{m, s}^\circ )\) are asymptotically negligible. This negligibility claim follows from the following facts:

  1. (A)

    \(Y_{i,m}=o_P(m)\), \(1\le i\le s\), where \(Y_{i,m}\) denotes the number of descendants of \(v_{i,m}^\circ \) in \(T_m^\circ \).

  2. (B)

    \(X_{i,m}=o_P(m)\), \(1\le i\le s\), where \(X_{i,m}\) denotes the number of descendants of \(u_{i,m}^\circ \) in \(T_m^\circ \) that are not in the subtree that contains \(v_{i,m}^\circ \).

  3. (C)

    For every \(\varepsilon >0\),

    $$\begin{aligned} {{\,\mathrm{{\mathbb {P}}}\,}}\big (\exists v\in T_m^\circ : v\text { has at least three subtrees in }T_m^\circ \text { each of size }\ge \varepsilon m\big ) \rightarrow 0\ \ \text { as }\ \ m\rightarrow \infty . \end{aligned}$$

(A) and (C) follow from (A.5) and the facts that \(q_{{\mathbf {e}}^\circ }(y_i^\circ )\) is almost surely a leaf in \({\mathscr {T}}_{{\mathbf {e}}^\circ }\) and that \({\mathscr {T}}_{{\mathbf {e}}^\circ }\) is almost surely binary. The proof of (B) is also routine. \(\square \)

1.2 Sketch of proof of Theorem 3.13

Assume that for each \(m\ge 1\), \({\varvec{k}}^{\scriptscriptstyle (m)}=(k_i^{\scriptscriptstyle (m)}, i\ge 0)\), where \(k_i^{\scriptscriptstyle (m)}\) are nonnegative integers satisfying \(\sum _{i\ge 0}k_i^{\scriptscriptstyle (m)}=m\) and \(\sum _{i\ge 0}ik_i^{\scriptscriptstyle (m)}=m-1\). Then there exist trees on m vertices in which for each \(i\ge 0\), there are exactly \(k_i^{\scriptscriptstyle (m)}\) many vertices with i many children. We call \({\varvec{k}}^{\scriptscriptstyle (m)}\) the child sequence of such a tree. Assumption 3.4 gives the criterion for graphs with given degree sequences to be critical. The following assumption gives the analogous criterion for plane trees with given child sequences.

Assumption A.4

There exists a pmf \((p_0, p_1,\ldots )\) with

$$\begin{aligned} p_0>0,\quad \sum _{i\ge 1}i p_i=1,\quad \text {and }\sum _{i\ge 1}i^2 p_i<\infty \end{aligned}$$

such that

$$\begin{aligned} \frac{k_i^{\scriptscriptstyle (m)}}{m}\rightarrow p_i\ \text { for }\ i\ge 0,\ \text { and }\ \frac{1}{m}\sum _{i\ge 0}i^2 k_i^{\scriptscriptstyle (m)}\rightarrow \sum _{i\ge 1}i^2 p_i. \end{aligned}$$

We will write \(\sigma ^2 = \sum _i i^2 p_i - 1\) for the variance associated with the pmf \((p_0, p_1,\ldots )\).

Let \({\mathbb {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}\) be the set of plane trees with child sequence \({\varvec{k}}^{\scriptscriptstyle (m)}\). Let \({\mathscr {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}\) be a uniform element of \({\mathbb {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}\) endowed with the tree distance and the uniform probability measure on m vertices and viewed as a metric measure space. Broutin and Marckert [34] showed that under Assumption A.4, \(\sigma m^{-1/2}{\mathscr {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}{\mathop {\longrightarrow }\limits ^{\text {d}}}{\mathscr {T}}_{2{\mathbf {e}}}\) w.r.t. GHP topology. The following variant of this result follows from [26, Lemma 7.4 and Lemma 7.6]:

Lemma A.5

Suppose \({\varvec{k}}^{\scriptscriptstyle (m)}\) satisfies Assumption A.4. Further, suppose \(f_m:\{0, 1,\ldots \}\rightarrow [0,1]\) is such that

$$\begin{aligned} \sum _{i\ge 0}k_i^{\scriptscriptstyle (m)} f_m(i)=1,\ \ \text { and }\ \ \lim _{m\rightarrow \infty }\ \max _{i: k_i^{\scriptscriptstyle (m)}>0}\ f_m(i)=0. \end{aligned}$$

Let \({\mathscr {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}^{f_m}\) be a uniform element of \({\mathbb {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}\) endowed with the tree distance and the measure that assigns probability \(f_m(i)\) to any node that has i children, \(i\ge 0\). Then

$$\begin{aligned} \sigma m^{-1/2}\cdot {\mathscr {T}}_{{\varvec{k}}^{\scriptscriptstyle (m)}}^{f_m}{\mathop {\longrightarrow }\limits ^{\text {d}}}{\mathscr {T}}_{2{\mathbf {e}}}\ \ \text { w.r.t. GHP topology}. \end{aligned}$$

Now we can prove Theorem 3.13 using the above lemma and the techniques used in the proof of [26, Theorem 2.2].

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Addario-Berry, L., Sen, S. Geometry of the minimal spanning tree of a random 3-regular graph. Probab. Theory Relat. Fields 180, 553–620 (2021). https://doi.org/10.1007/s00440-021-01071-3

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  • DOI: https://doi.org/10.1007/s00440-021-01071-3

Keywords

  • Minimal spanning tree
  • Gromov–Hausdorff distance
  • Critical percolation
  • Real tree
  • Random regular graphs
  • Graphs with prescribed degree sequence
  • Configuration model

Mathematics Subject Classification

  • Primary 60C05
  • 05C80