Proof. Recall that $\mathbb{I}$ denotes the intersection $\mathbb{G}_{n,\lambda } \cap \mathbb{G}^\varepsilon _{n,\lambda }$ . The graph $\mathbb{I}$ is a $\mathcal G(n,\theta )$ random graph, where

\begin{equation*}\theta \;:\!=\;p(1-\varepsilon + \varepsilon p)=\frac {1-\varepsilon (1+o(1))}{n}.\end{equation*}

Fix some edge $e=\{u,v\}$ in $K_n$ . We consider two disjoint possibilities for the occurrence of the event $e\in \mathbb{J}$ :

1. 1. The event $A=A_e=\{e\in \mathrm{conn}(\mathbb{I})\}$ where $e$ belongs to a cycle that is contained in both graphs, or

2. 2. the event $B=B_e=\{e\in \mathbb{J}\setminus \mathrm{conn}(\mathbb{I})\}$ where there are two distinct cycles in $\mathbb{G}_{n,\lambda }$ and $\mathbb{G}^\varepsilon _{n,\lambda }$ both containing $e$ , and there is no cycle in $\mathbb{I}$ containing $e.$

We bound the probability of $A$ by observing it occurs if and only if $e\in \mathbb{I}$ and there is a path in the graph $\mathbb{I} - e$ from $v$ to $u$ . By enumerating all the paths from $v$ to $u$ with $k\ge 1$ additional vertices we find that

(3.1) \begin{equation} \mathbb{P}(A) \le \theta \sum _{k\ge 1}n^k\theta ^{k+1} \le \frac{\theta ^3 n}{1-\theta n}= \frac{1+o(1)}{\varepsilon n^2}, \end{equation}

where the last inequality follows from the relations $\theta \le 1/n$ and $1-\theta n=\varepsilon (1+o(1))$ .

Figure 1. The three combinations of internal and external paths between $u$ and $v$ that can cause the occurrence of $B$ .

Next, we turn to bound the probability of $B$ . Let $C_x$ , for $x\in \{u,v\}$ , denote the component of the vertex $x$ in the graph $\mathbb{I}-e$ . We further denote $\mathbb K_1\;:\!=\;\mathbb{G}_{n,\lambda }\setminus (C_u\cup C_v)$ and $\mathbb K_2\;:\!=\;\mathbb{G}^\varepsilon _{n,\lambda }\setminus (C_u\cup C_v)$ .

Claim 3.2. For every $C_u,C_v, \mathbb K_1,\mathbb K_2$ as above there holds

\begin{equation*} \mathbb {P}(B\,|\,C_u,C_v,\mathbb K_1,\mathbb K_2) \le \textbf {1}_{C_u\ne C_v}\cdot \theta \cdot (|C_u||C_v|)^2\cdot \prod _{i=1}^{2}\left (\rho +\rho ^2\sum _{j\ge 1}|C_j(\mathbb K_i)|^2\right )\,, \end{equation*}

where $ \rho \;:\!=\; p\varepsilon (1-p)/(1-\theta ).$

Proof. We first note that $C_u$ is either equal or disjoint to $C_v$ , and that in the former case there exists a path from $v$ to $u$ in $\mathbb{I} - e$ . We observe that if $C_u=C_v$ then the event $B$ does not occur, hence both sides in the claimed inequality are equal to $0$ . Indeed, this is derived directly by combining the facts $B\subseteq \{e\in \mathbb{I}\}\cap A^c$ and $A=\{e\in \mathbb{I}\}\cap \{C_u=C_v\}.$

Suppose that $C_u\cap C_v=\emptyset$ , and consider the edge sets

\begin{equation*}F_0\;:\!=\;\{\{a,b\}\;:\;a\in C_u,\,b\in C_v\}\setminus \{e\},\end{equation*}

and

\begin{equation*}F_1\;:\!=\;\{\{a,b\}\;:\;a\in C_u\cup C_v, b\notin C_u\cup C_v\}.\end{equation*}

Note that for every $f\in F_0\cup F_1$ , the only information that is exposed by our conditioning is that $f\notin \mathbb{I}$ . Therefore, for every two edge subsets $L_1,L_2\subset F_0\cup F_1$ there holds

(3.2) \begin{equation} \mathbb{P}(L_1\subseteq \mathbb{G}_{n,\lambda },L_2\subseteq \mathbb{G}^\varepsilon _{n,\lambda }\,|\,C_u,C_v) \le \rho ^{|L_1|+|L_2|} \end{equation}

Indeed, if $L_1\cap L_2\ne \emptyset$ then this conditional probability is $0$ since no edge of $F_0\cup F_1$ is in $\mathbb{I}$ . Otherwise, by the independence between the different edges, (3.2) follows from the fact that, for every edge $f$ , $ \mathbb{P}(f\in \mathbb{G}_{n,\lambda }\,|\,f\notin \mathbb{I})=\mathbb{P}(f\in \mathbb{G}^\varepsilon _{n,\lambda }\,|\,f\notin \mathbb{I})=\rho .$

We consider two different partitions of $F_1$ given by

\begin{equation*} F_1 = \bigcup _{j\ge 1,x\in \{u,v\}} F_{x,j,i},\,\,\,\,i=1,2\,, \end{equation*}

where $F_{x,j,i}$ consists of all the edges between $C_x$ and the $j$ -th largest connected component $C_j(\mathbb{K}_i)$ of the graph $\mathbb{K}_i$ . A path from $v$ to $u$ in $\mathbb{G}_{n,\lambda } - e$ can either be internal and involve an edge from $F_0$ , or be external and involve one edge from $F_{v,j,1}$ and one from $F_{u,j,1}$ for some $j\ge 1$ , using the edges from $C_j(\mathbb{K}_1)$ to complete the path. Clearly, a similar statement holds for $\mathbb{G}^\varepsilon _{n,\lambda }$ where $F_{x,j,1}$ is replaced by $F_{x,j,2}$ for both $x\in \{u,v\}$ (See Fig. 1). Therefore, we claim that

(3.3) \begin{align} \nonumber & \mathbb{P}(B\,|\,C_v,C_u,\mathbb{K}_1,\mathbb{K}_2) \le \\ \nonumber & \quad \le \, \textbf{1}_{C_u\ne C_v}\cdot \theta \cdot \left (\rho ^2|F_0|^2 +\rho ^3|F_0|\sum _{i=1}^{2}\sum _{j\ge 1}\ |F_{v,j,i}||F_{u,j,i}| +\rho ^4\prod _{i=1}^{2}\sum _{j\ge 1}\ |F_{u,j,i}||F_{v,j,i}|\right ) \\ & \quad= \textbf{1}_{C_u\ne C_v}\cdot \theta \cdot \prod _{i=1}^{2}\left (\rho |F_0|+\rho ^2\sum _{j\ge 1}\ |F_{u,j,i}||F_{v,j,i}|\right ). \end{align}

Indeed, every term in the second line corresponds to a different combination of internal and external paths. The first term corresponds to having two internal paths so we have $|F_0|^2$ choices for having an edge from $F_0$ in both graphs, and the probability that the two edges actually appear is at most $\rho ^2$ by (3.2). Similarly, the second term accounts for having one internal and one external path, where for the external path, say in $\mathbb{G}_{n,\lambda }$ , we need to choose the component $C_j(\mathbb{K}_1)$ we use, as well as an edge from $F_{u,j,1}$ and an edge from $F_{v,j,1}$ . We multiply by $\rho ^3\cdot |F_0|$ , since in addition to having these two edges appear in $\mathbb{G}_{n,\lambda }$ , we also choose an edge from $F_0$ to appear in $\mathbb{G}^\varepsilon _{n,\lambda }$ . The last term is derived by considering the case of two external paths, as we need to choose, for both graphs $ \mathbb{K}_i$ , a component $C_j( \mathbb{K}_i)$ , and edges from $F_{v,j,i}$ and $F_{u,j,i}.$ To conclude, note the multiplicative term $\theta$ accounting for the event $e\in \mathbb{I}$ . Alternatively, (3.3) can be understood as letting each of the graphs $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ either choose an internal path with a cost of $\rho$ or an external path with a cost of $\rho ^2$ . The product of these two terms appears due to the negative correlations from (3.2). The claim is derived from (3.3) by noting that $|F_0|\lt |C_u||C_v|$ , $|F_{x,j,i}|=|C_x||C_j(\mathbb{K}_i)|$ for every $x,j$ and $i$ , and a straightforward manipulation.

We proceed by observing that

(3.4) \begin{equation} \sum _{j\ge 1}|C_j(\mathbb K_1)|^2\le \sum _{j\ge 1}|C_j(\mathbb{G}_{n,\lambda })|^2\,\mbox{ and }\, \sum _{j\ge 1}|C_j(\mathbb K_2)|^2 \le \sum _{j\ge 1}|C_j(\mathbb{G}^\varepsilon _{n,\lambda })|^2\,, \end{equation}

since $\mathbb K_1,\mathbb K_2$ are subgraphs of $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ respectively.

Next, for a positive $c\in{\mathbb{R}}$ , denote by $E_c$ the event that

\begin{equation*} \max \left \{\sum _{j\ge 1}|C_j(\mathbb {G}_{n,\lambda })|^2,\sum _{j\ge 1}|C_j(\mathbb {G}^\varepsilon _{n,\lambda })|^2\right \} \le cn^{4/3}, \end{equation*}

and recall that [Reference Lubetzky and Peled18, Theorem 1], [Reference Frilet11] establish that if $\varepsilon ^3n\to \infty$ then the pair

\begin{equation*} n^{-2/3}\cdot \left ( (|C_j(\mathbb {G}_{n,\lambda })|)_{j\ge 1}, (|C_j(\mathbb {G}^\varepsilon _{n,\lambda })|)_{j\ge 1} \right ) \end{equation*}

weakly converges in $\ell _2$ to a pair of independent copies of a random sequence whose law was identified by Aldous [Reference Aldous5]. Therefore,

(3.5) \begin{equation} \lim _{c\to \infty }\lim _{n\to \infty }\mathbb{P}(E_c)=1. \end{equation}

By combining (3.4) and Claim 3.2 we find that

(3.6) \begin{equation} \mathbb{P}\left ( B,E_c \,|\,C_v,C_u\right ) \leq \textbf{1}_{C_u\ne C_v}\cdot \theta \cdot (|C_u||C_v|)^2\cdot \left (\rho +c\cdot \rho ^2 n^{4/3}\right )^2\,. \end{equation}

Let $Y$ denote the size of the connected component of a fixed vertex in a $\mathcal{G}(n,\theta )$ random graph. Note that for every choice of $C_u$ , the random variable $\textbf{1}_{C_u\ne C_v}|C_v|^2$ is stochastically bounded from above by $Y^2$ . Indeed, If $v\in C_u$ then $\textbf{1}_{C_u\ne C_v}=0$ . Otherwise, $C_v$ is the component of $v$ in the $\mathcal G(n-|C_u|,\theta )$ random graph $\mathbb{I}\setminus C_u$ . As a result, $|C_v|$ is indeed dominated by $Y$ . Therefore,

(3.7) \begin{align} \nonumber \mathbb{E}[\textbf{1}_{C_u\ne C_v}(|C_u||C_v|)^2] =\, &\mathbb{E}\left [|C_u|^2\cdot \mathbb{E}\left [ \left . \textbf{1}_{C_u\ne C_v}|C_v|^2\,\right |\,C_u\right ]\right ]\\ \le \,& \mathbb{E}\left [|C_u|^2 \right ]\mathbb{E}[Y^2] \nonumber \\ \le \,& \mathbb{E}[Y^2]^2. \end{align}

In addition,

(3.8) \begin{equation} \mathbb{E}[Y^2] = \frac 1n\mathbb{E}_{G\sim \mathcal G(n,\theta )}\left [ \sum _{j\ge 1} |C_j(G)|^{3} \right ]\le \frac{1}{(1-n\theta )^{3}} = \frac{1+o(1)}{\varepsilon ^3}, \end{equation}

where the first equality is derived by averaging over the vertices and accounting for the contribution of each connected component, the inequality follows from the work of Janson and Łuczak on subcritical random graphs [Reference Janson and Luczak15], and the second equality by $1-n\theta = (1-o(1))\varepsilon$ . By assigning (3.7), (3.8), and the relations $\theta \lt 1/n$ and $\rho = (1+o(1))\varepsilon/n$ in (3.6), we find that

(3.9) \begin{equation} \mathbb{P}(B,E_c) \le \frac{1+o(1)}{\varepsilon ^6 n}\cdot \left (\frac{\varepsilon }{n} + \frac{c\varepsilon ^2}{n^{2/3}} \right )^2 = \frac{1+o(1)}{\varepsilon n^2}\left ( (\varepsilon ^3 n)^{-1/2} + c(\varepsilon ^3 n)^{-1/6} \right )^2. \end{equation}

Therefore, we derive from (3.1), (3.9) and $\varepsilon ^3n\to \infty$ that

\begin{equation*} \mathbb {E}[|\mathbb {J}|\cdot \textbf {1}_{E_c}] = \binom n2\mathbb {P}(e\in \mathbb {J},E_c)\le \frac {n^2}{2}(\mathbb {P}(A) + \mathbb {P}(B,E_c))\le \frac {1+o(1)}{2\varepsilon }. \end{equation*}

Finally, note that by Markov’s inequality,

\begin{align*} \mathbb{P}(|\mathbb{J}|\gt \omega \varepsilon ^{-1}) \le &\, \mathbb{P}(E_c^c) + \mathbb{P}(|\mathbb{J}|\cdot \textbf{1}_{E_c}\gt \omega \varepsilon ^{-1})\\ \le &\,1-\mathbb{P}(E_c)+\frac{1+o(1)}{2\omega }. \end{align*}

This concludes the proof using (3.5) and the assumption that $\omega \to \infty$ as $n\to \infty$ .

Proof. Recall that $\mathbb{J} =\mathrm{conn}(\mathbb{G}_{n,\lambda })\cap \mathrm{conn}(\mathbb{G}^\varepsilon _{n,\lambda })$ , and $\check{\mathbb{I}}=\{e\in K_n\,:\,w_e\le p,b_e=0\}$ is the random graph consists of the edges in $\mathbb{G}_{n,\lambda }\cap \mathbb{G}^\varepsilon _{n,\lambda }$ whose weight had not been resampled. We sample the graphs $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda },{\mathbb{M}}_{n,\lambda },{\mathbb{M}}^\varepsilon _{n,\lambda }$ using $W_n,W_n^\varepsilon$ (See section 2), and set $\mathbb{F}_{n,\lambda }\;:\!=\;{\mathbb{M}}_{n,\lambda }$ . In addition, let $\mathbb{F}^\varepsilon _{n,\lambda }$ be the MSF of $\mathbb{G}^\varepsilon _{n,\lambda }$ endowed with the following edge weights:

\begin{equation*} \tilde w_e = \begin {cases} w_e^\varepsilon & e\in \mathbb {G}^\varepsilon _{n,\lambda }\setminus (\check {\mathbb {I}}\cap \mathbb {J}), \\ p\cdot w^{\prime}_e & e\in \check {\mathbb {I}}\cap \mathbb {J}, \end {cases} \end{equation*}

where $w^{\prime}_e$ is an independent $\mathrm{U}[0,1]$ variable. First, we claim that the forests $\mathbb{F}_{n,\lambda },\mathbb{F}^\varepsilon _{n,\lambda }$ are retained respectively from $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ by independent cycle-breaking algorithms. Namely, conditioned on $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ , the pair $(\mathbb{F}_{n,\lambda },\mathbb{F}^\varepsilon _{n,\lambda })$ is $\mathcal{K}^{\infty }(\mathbb{G}_{n,\lambda })\times \mathcal{K}^{\infty }(\mathbb{G}^\varepsilon _{n,\lambda })$ -distributed. This follows from the fact that conditioned on $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ and $\check{\mathbb{I}}$ , the weights

\begin{equation*} (w_e)_{e\in \mathrm {conn}(\mathbb {G}_{n,\lambda })} \mbox { and }(\tilde w_e)_{e\in \mathrm {conn}(\mathbb {G}^\varepsilon _{n,\lambda })},\end{equation*}

which determine the edges that are removed in the cycle-breaking algorithms, are i.i.d. Indeed, the only dependency between weights can occur via an edge from $\mathbb{J}$ but for every such an edge $e$ , the weights in both graphs are independent either due to resampling (if $e\notin \check{\mathbb{I}}$ ) or by the definition of $\tilde w_e$ (if $e\in \check{\mathbb{I}}$ ).

Next, we bound the distance $d_{\textrm{GHP}}^4(M_{n,\lambda }^\varepsilon,F_{n,\lambda }^\varepsilon ).$ Denote by $B_j,\,j\ge 1$ , the event that the trees $C_j({\mathbb{M}}^\varepsilon _{n,\lambda })$ and $C_j(\mathbb{F}^\varepsilon _{n,\lambda })$ are different. Note that the forests ${\mathbb{M}}^\varepsilon _{n,\lambda }$ and $\mathbb{F}^\varepsilon _{n,\lambda }$ are retained from $\mathbb{G}^\varepsilon _{n,\lambda }$ by the cycle-breaking algorithm using, respectively, the edge weights $(w_e^\varepsilon )_{e\in \mathbb{G}^\varepsilon _{n,\lambda }}$ and $(\tilde w_e)_{e\in \mathbb{G}^\varepsilon _{n,\lambda }}$ , which differ only on $\check{\mathbb{I}}\cap \mathbb{J}$ . Therefore, if $B_j$ occurs then there exists a cycle $\gamma$ in $C_j(\mathbb{G}^\varepsilon _{n,\lambda })$ and an edge $f\in \gamma \cap \check{\mathbb{I}}\cap \mathbb{J}$ that is the heaviest in $\gamma$ with respect to one of the edge weights. Otherwise, the two runs of the cycle-breaking algorithms on $C_j(\mathbb{G}^\varepsilon _{n,\lambda })$ must be identical.

Let $S$ denote the number of distinct simple cycles in $C_j(\mathbb{G}^\varepsilon _{n,\lambda })$ , $R$ the length of the shortest cycle in $C_j(\mathbb{G}^\varepsilon _{n,\lambda })$ (or $R=\infty$ if the component is acyclic), and let $\gamma$ be a cycle in $C_j(\mathbb{G}^\varepsilon _{n,\lambda })$ . Conditioned on $\mathbb{G}^\varepsilon _{n,\lambda }$ and $\mathbb{J}$ , the probability that the heaviest edge of $\gamma$ (in each of the weights) belongs to $\mathbb{J}$ is bounded from above by $|\mathbb{J}|/{R}$ , since $|\gamma |$ is bounded from below by $R$ . Hence, by taking the union bound over all the cycles in the component and the two edge weights we find that $\mathbb{P}(B_j\,|\,\mathbb{G}^\varepsilon _{n,\lambda },\mathbb{J}) \leq 2 \cdot S \cdot |\mathbb{J}|/R.$ Therefore, for every $\omega \gt 0$ , the probability of $B_j$ conditioned on the event $C$ that $|\mathbb{J}|\lt \omega \varepsilon ^{-1},\,S\lt \omega,$ and $R \gt n^{1/3}\omega ^{-1}$ , is bounded by

\begin{equation*} \mathbb {P}\left (B_j\,|\,C\right ) \leq \frac {2\cdot \omega \cdot (\omega \varepsilon ^{-1}) }{n^{1/3}\omega ^{-1}}. \end{equation*}

Consequently,

(3.11) \begin{equation} \mathbb{P}(B_j) \le \mathbb{P}(|\mathbb{J}|\ge \omega \varepsilon ^{-1}) + \mathbb{P}(S\ge \omega ) + \mathbb{P}(R\le n^{1/3}\omega ^{-1}) + \frac{2\cdot \omega \cdot (\omega \varepsilon ^{-1}) }{n^{1/3}\omega ^{-1}}. \end{equation}

Suppose that $\omega =\omega (n)\to \infty$ as $n\to \infty$ . Lemma 3.1 asserts that first term in (3.11) is negligible. In addition, the second and third terms are also negligible by known results on critical random graphs. Namely, $S$ converges in distribution to an almost-surely finite limit by [Reference Aldous5, Reference Luczak, Pittel and Wierman20], and the fact that $n^{-1/3}\omega R$ almost surely diverges follows from [Reference Addario-Berry, Broutin and Goldschmidt3] for unicyclic components, and from [Reference Luczak, Pittel and Wierman20] for complex components (components with more than one cycle). Choosing $\omega =\omega (n)$ such that $\omega \to \infty$ and $\omega ^3/ (\varepsilon n^{1/3})\to 0$ as $n\to \infty$ results in $\mathbb{P}(B_j)\to 0$ , for every $j\geq 1$ .

To complete the proof, observe that for every $\eta \gt 0$ and $N\ge 1$ there holds

\begin{equation*} \mathbb{P}(d_{\textrm{GHP}}^4(M_{n,\lambda }^\varepsilon, F_{n,\lambda }^\varepsilon )\gt \eta ) \le \sum _{j=1}^{N-1} \mathbb{P}(B_j) + \mathbb{P}\left ( \sum _{j=N}^\infty d_{\textrm{GHP}}(M_{n,\lambda,j}^\varepsilon, F_{n,\lambda,j}^\varepsilon )^4 \gt \eta \right ). \end{equation*}

The first sum is negligible as $n\to \infty$ since $\mathbb{P}(B_j)\to 0$ for every $j\ge 1$ . In addition, by the fact that both $M_{n,\lambda }^\varepsilon$ and $F_{n,\lambda }^\varepsilon$ converge in distribution as $n\to \infty$ in $(\mathbb L_4,d_{\textrm{GHP}}^4)$ we have that

\begin{equation*} \lim _{N\to \infty }\limsup _{n\to \infty } \mathbb {P}\left ( \sum _{j=N}^\infty d_{\textrm {GHP}}(M_{n,\lambda,j}^\varepsilon, F_{n,\lambda,j}^\varepsilon )^4 \gt \eta \right )= 0, \end{equation*}

which completes the proof of the lemma.

Proof. We start by describing, in very high-level terms, how the space $\mathscr{M}_\lambda$ is constructed. Recall the random measured metric space $\mathscr{G}_\lambda$ that was introduced in refs. [Reference Addario-Berry, Broutin and Goldschmidt2], [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4], and was shown to be the limit of $G_{n,\lambda }$ in distribution, as $n\to \infty$ , in $(\mathbb{L}_4,d_{\textrm{GHP}}^4).$ The random space $\mathscr{M}_\lambda$ was defined conditioned on $\mathscr{G}_\lambda$ as being $\mathcal{K}^{\infty }(\mathscr{G}_\lambda )$ -distributed, where $\mathcal{K}^{\infty }$ is the continuous analog of the cycle-breaking algorithm.

Next, denote by $(s(G_{n,\lambda,i}))_{i\geq 1}$ and $(s(\mathscr{G}_{\lambda,i}))_{i\geq 1}$ the sequence of surpluses of the components in $G_{n,\lambda }$ and $\mathscr{G}_{\lambda }$ respectively. In addition, let $(r(G_{n,\lambda,i}))_{i\geq 1}$ and $(r(\mathscr{G}_{\lambda,i}))_{i\geq 1}$ be the sequence of minimal length of a core edge in each component. We refer the reader to [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4] for precise definitions. The following claim follows from the proof of [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4, Theorem 4.4].

Claim 3.5. Let $\Omega$ be a probability space in which $\mathbb{G}_{n,\lambda },\mathscr{G}_{\lambda }$ are commonly defined such that $\Omega$ -almost-surely there holds that

\begin{align*} G_{n,\lambda }&\to \mathscr{G}_{\lambda }\mbox{ in }(\mathbb{L}_4,d_{\textrm{GHP}}^4),\\ (s(G_{n,\lambda,i}))_{i\geq 1}&\to (s(\mathscr{G}_{\lambda,i}))_{i\geq 1},\\ (r(G_{n,\lambda,i}))_{i\geq 1}&\to (r(\mathscr{G}_{\lambda,i}))_{i\geq 1}, \end{align*}

as $n\to \infty$ . Then, for every continuity set $S$ of $(\mathbb{L}_4,d_{\textrm{GHP}}^4)$ for $\mathscr{M}_\lambda$ , the convergence

\begin{equation*} \mathbb {P}(M_{n,\lambda } \in S \,|\,\mathbb {G}_{n,\lambda }) \to \mathbb {P}(\mathscr {M}_{\lambda } \in S \,|\,\mathscr {G}_\lambda )\,,\,\mbox {as $n\to \infty $}, \end{equation*}

of random variables occurs $\Omega$ -almost surely. Here, conditioned on $\mathbb{G}_{n,\lambda }$ and $\mathscr{G}_{\lambda }$ , ${\mathbb{M}}_{n,\lambda }$ is $\mathcal{K}^{\infty }(\mathbb{G}_{n,\lambda })$ -distributed, $M_{n,\lambda }\;:\!=\;{\mathcal{S}}({\mathbb{M}}_{n,\lambda })$ and $\mathscr{M}_\lambda$ is $\mathcal{K}^{\infty }(\mathscr{G}_\lambda )$ -distributed.

In fact, it is proved in ref. [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4, Theorem 4.4] that under the conditions of Claim 3.5, the cycle-breaking algorithms carried out on $\mathbb{G}_{n,\lambda },\mathscr{G}_\lambda$ can be coupled such that the convergence of $M_{n,\lambda }\to \mathscr{M}_{\lambda }$ in $(\mathbb{L}_4,d_{\textrm{GHP}}^4)$ also occurs $\Omega$ -almost-surely.

Back to noise sensitivity, the results in refs. [Reference Lubetzky and Peled18, Theorem 2] and [Reference Frilet11, Theorem 9.1.1], establish that if $\varepsilon ^3n\to \infty$ as $n\to \infty$ then

(3.12) \begin{align} (G_{n,\lambda },G_{n,\lambda }^\varepsilon ) & \xrightarrow{\,\mathrm{d}\,} (\mathscr{G}_\lambda,\mathscr{G}^{\;\prime}_\lambda),\,\mbox{and} \end{align}

(3.13) \begin{align} \left ((s(G_{n,\lambda,i}))_{i\geq 1},(s(G^\varepsilon _{n,\lambda,i}))_{i\geq 1}\right )& \xrightarrow{\,\mathrm{d}\,}\left ((s(\mathscr{G}_{\lambda,i}))_{i\geq 1},(s(\mathscr{G}^{\;\prime}_{\lambda,i}))_{i\geq 1}\right ), \end{align}

where $\mathscr{G}^{\;\prime}_\lambda$ is an independent copy of $\mathscr{G}_\lambda$ . Here the first convergence is in $(\mathbb L_4,d_{\textrm{GHP}}^4)$ , and the second in the sense of finite dimensional distributions,

The proof of [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4, Theorem 4.1] shows that this convergence can be extended to the minimal lengths of core edges, implying that

(3.14) \begin{align} \left ((r(G_{n,\lambda,i}))_{i\geq 1},(r(G^\varepsilon _{n,\lambda,i}))_{i\geq 1}\right )& \xrightarrow{\,\mathrm{d}\,}\left ((r(\mathscr{G}_{\lambda,i}))_{i\geq 1},(r(\mathscr{G}^{\;\prime}_{\lambda,i}))_{i\geq 1}\right ), \end{align}

as $n\to \infty$ .

Using Skorohod’s representation theorem, we may work in a probability space $\Omega$ in which the convergences. Equations (3.12),(3.13) and (3.14) occur almost surely. In addition, we can consider the distributions of $F_{n,\lambda },F_{n,\lambda }^\varepsilon,\mathscr{M}_\lambda$ and its independent copy $\mathscr{M}^{\;\prime}_\lambda$ by constructing them via $\Omega$ . Namely, conditioned on $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda },\mathscr{G}_\lambda,\mathscr{G}^{\;\prime}_\lambda$ sampled in $\Omega$ , we consider the (distributions of the) following random elements:

• • The pair $(\mathbb{F}_{n,\lambda },\mathbb{F}^\varepsilon _{n,\lambda })$ is $\mathcal{K}^{\infty }(\mathbb{G}_{n,\lambda })\times \mathcal{K}^{\infty }(\mathbb{G}^\varepsilon _{n,\lambda })$ -distributed, $F_{n,\lambda }\;:\!=\;{\mathcal{S}}(\mathbb{F}_{n,\lambda })$ and $F_{n,\lambda }^\varepsilon ={\mathcal{S}}(\mathbb{F}^\varepsilon _{n,\lambda })$ , and

• • The pair $(\mathscr{M}_\lambda,\mathscr{M}^{\;\prime}_\lambda)$ is $\mathcal{K}^{\infty }(\mathscr{G}_\lambda ) \times \mathcal{K}^{\infty }(\mathscr{G}^{\;\prime}_\lambda)$ -distributed.

We observe that the pair $(F_{n,\lambda },F_{n,\lambda }^\varepsilon )$ is conditionally independent given $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ , and the pair $(\mathscr{M}_\lambda,\mathscr{M}^{\;\prime}_\lambda)$ is independent and identically distributed.

Our goal is to show that for every continuity sets $S,S'$ of $(\mathbb L_4,d_{\textrm{GHP}}^4)$ for the distribution of $\mathscr{M}_\lambda$ there holds

\begin{equation*} \mathbb {P}(F_{n,\lambda }\in S,F^\varepsilon _{n,\lambda }\in S') \to \mathbb {P}(\mathscr {M}_\lambda \in S)\mathbb {P}(\mathscr {M}_\lambda '\in S') \end{equation*}

as $n\to \infty$ .

Note that by our assumption on the almost-sure convergences in $\Omega$ , we can apply Claim 3.5 twice and obtain that for every two such continuity sets $S,S'$ there holds that both convergences

(3.15) \begin{equation} \mathbb{P}(F_{n,\lambda }\in S \,|\,\mathbb{G}_{n,\lambda }) \to \mathbb{P}(\mathscr{M}_\lambda \in S \,|\,\mathscr{G}_\lambda )\,,\,\mbox{as $n\to \infty $}, \end{equation}

and

(3.16) \begin{equation} \mathbb{P}(F_{n,\lambda }^\varepsilon \in S' \,|\,\mathbb{G}^\varepsilon _{n,\lambda }) \to \mathbb{P}(\mathscr{M}^{\;\prime}_\lambda\in S' \,|\,\mathscr{G}^{\;\prime}_\lambda)\,,\,\mbox{as $n\to \infty $} \end{equation}

occur $\Omega$ -almost surely. Consequently, the proof is concluded as follows:

\begin{align*} \mathbb{P}(F_{n,\lambda }\in S,F^\varepsilon _{n,\lambda }\in S')&=\, \mathbb{E}[\mathbb{P}(F_{n,\lambda }\in S,F^\varepsilon _{n,\lambda }\in S'\,|\,\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda })]\\ &=\, \mathbb{E}[\mathbb{P}(F_{n,\lambda }\in S \,|\,\mathbb{G}_{n,\lambda })\cdot \mathbb{P}(F_{n,\lambda }^\varepsilon \in S' \,|\,\mathbb{G}^\varepsilon _{n,\lambda })]\\ &\to \, \mathbb{E}[\mathbb{P}(\mathscr{M}_\lambda \in S \,|\,\mathscr{G}_\lambda )\cdot \mathbb{P}(\mathscr{M}^{\;\prime}_\lambda\in S' \,|\,\mathscr{G}^{\;\prime}_\lambda)]\\ &=\, \mathbb{E}[\mathbb{P}(\mathscr{M}_\lambda \in S \,|\,\mathscr{G}_\lambda )]\cdot \mathbb{E}[\mathbb{P}(\mathscr{M}^{\;\prime}_\lambda\in S' \,|\,\mathscr{G}^{\;\prime}_\lambda)]\\ &=\,\mathbb{P}(\mathscr{M}_\lambda \in S)\cdot \mathbb{P}(\mathscr{M}^{\;\prime}_\lambda\in S'). \end{align*}

The first equality holds by the law of total expectation, and the second equality is due to the conditional independence of $F_{n,\lambda },F_{n,\lambda }^\varepsilon$ given $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ . The convergence, which occurs as $n\to \infty$ , is obtained from the $\Omega$ -almost-sure convergences (3.15), (3.16) and using the dominated convergence theorem. The next equality is obtained by the independence of $\mathscr{G}_\lambda$ , $\mathscr{G}^{\;\prime}_\lambda$ which is where the noise sensitivity of the measured metric structure of $\mathbb{G}_{n,\lambda },\mathbb{G}^\varepsilon _{n,\lambda }$ is being used. The last equality follows from the law of total expectation.

Proof of Theorem 1.2, Part (1). Denote the polish metric space $S=(\mathbb L_4,d_{\textrm{GHP}}^4)^2$ endowed with some product metric $\rho$ . Suppose that the random elements

\begin{equation*}((M_{n,\lambda },M_{n,\lambda }^\varepsilon ),(F_{n,\lambda },F_{n,\lambda }^\varepsilon )) \in S\times S\end{equation*}

are sampled via the coupling from Lemma 3.3. Lemma 3.4 asserts that $(F_{n,\lambda },F_{n,\lambda }^\varepsilon )$ converges in distribution to a pair of independent copies of $\mathscr{M}_{\lambda }$ . In addition, by Lemma 3.3,

\begin{equation*} \rho ((M_{n,\lambda },M_{n,\lambda }^\varepsilon ),(F_{n,\lambda },F_{n,\lambda }^\varepsilon ))\xrightarrow {\,\mathrm {p}\,} 0\,,\end{equation*}

as $n\to \infty$ . Consequently, we derive from Theorem 3.6 that $(M_{n,\lambda },M_{n,\lambda }^\varepsilon )$ converges in distribution to a pair of independent copies of $\mathscr{M}_{\lambda }$ , as claimed.

Proof. We observe that conditioned on $\mathbb{G}_{n,\lambda }$ , the graph $\check{\mathbb{I}}$ is obtained from $\mathbb{G}_{n,\lambda }$ by removing each edge independently with probability $\varepsilon$ . Therefore, by [Reference Lubetzky and Peled18, Lemma 5.4], the event $A_j$ that $C_j(\check{\mathbb{I}})\subseteq C_j(\mathbb{G}_{n,\lambda })$ occurs with probability tending to $1$ as $n\to \infty .$ In addition, under the event $A_j$ , the event $B_j$ does not occur only if there exists an edge $e\in \mathrm{conn}(C_j(\mathbb{G}_{n,\lambda }))$ that $\check{\mathbb{I}}$ did not retain. Therefore, for every $\omega =\omega (n)\gt 0$ ,

(3.17) \begin{equation} \mathbb{P}(B_j^c\,|\,A_j) \le \mathbb{P}(|\mathrm{conn}(C_j(\mathbb{G}_{n,\lambda }))|\gt \omega n^{1/3}) + \varepsilon \omega n^{1/3}, \end{equation}

where the second term bounds the expected number of edges from $\mathrm{conn}(\mathbb{G}_{n,\lambda })$ that $\check{\mathbb{I}}$ did not retain, conditioned on $|\mathrm{conn}(C_j(\mathbb{G}_{n,\lambda }))|\le \omega n^{1/3}$ . We derive the claim by combining $\mathbb{P}(A_j)\to 1$ and assigning in (3.17) a sequence $\omega = \omega (n)$ such that $\omega \to \infty$ and $\varepsilon \omega n^{1/3}\to 0$ as $n\to \infty$ . Indeed, in such a case we have that $\mathbb{P}(n^{-1/3}|\mathrm{conn}(C_j(\mathbb{G}_{n,\lambda }))|\gt \omega )\to 0$ , since the maximum number of cycles in $\mathbb{G}_{n,\lambda }$ is bounded in probability [Reference Luczak, Pittel and Wierman20], and so is the length of the largest cycle in $C_j(\mathbb{G}_{n,\lambda })$ divided by $n^{1/3}$ [Reference Addario-Berry, Broutin and Goldschmidt3, Reference Luczak, Pittel and Wierman20].

Proof of Theorem 1.2 Part 2. Denote by $\check{\mathbb{M}}$ the MSF of the graph $\check{\mathbb{I}}$ endowed with the edge weights from $W_n$ , and let $\check M\;:\!=\;{\mathcal{S}}(\check{\mathbb{M}})$ . First, we argue that for every fixed $j \ge 1$ ,

(3.18) \begin{equation} d_{\textrm{GHP}}(M_{n,\lambda,j},\check M_j) \xrightarrow{\,\mathrm{p}\,} 0 \end{equation}

as $n\to \infty$ . By Claim 3.7, we can condition on the event $B_j$ . Under this event, the joint cycle-breaking algorithm running on $C_j(\mathbb{G}_{n,\lambda })$ and $C_j(\check{\mathbb{I}})$ removes the same edges in both graphs. Since $\check{\mathbb{I}}$ is a subgraph of $\mathbb{G}_{n,\lambda }$ , we deduce that $C_j({\mathbb{M}}_{n,\lambda })$ is obtained from $C_j(\check{\mathbb{M}})$ by the addition of the forest $C_j(\mathbb{G}_{n,\lambda }) \setminus C_j(\check{\mathbb{I}})$ . We derive (3.18) by the proof of [Reference Lubetzky and Peled18, Theorem 2], which shows that with probability tending to $1$ as $n\to \infty$ , the graph $C_j(\mathbb{G}_{n,\lambda })$ is contained in a neighbourhood of radius $o(n^{1/3})$ around $C_j(\check{\mathbb{I}})$ , and that the two graphs differ by $o(n^{2/3})$ vertices.

Since $(\mathbb{G}_{n,\lambda },\check{\mathbb{I}},W_n)\stackrel{d}{=}(\mathbb{G}^\varepsilon _{n,\lambda },\check{\mathbb{I}},W_n^\varepsilon )$ , we can use the same argument for $\mathbb{G}^\varepsilon _{n,\lambda }$ instead of $\mathbb{G}_{n,\lambda }$ , and find that

\begin{equation*} d_{\textrm {GHP}}(M_{n, \lambda,j},M_{n, \lambda,j}^\varepsilon ) \xrightarrow {\,\mathrm {p}\,} 0, \end{equation*}

as $n\to \infty$ . To conclude Theorem 1.2 Part 2 we need to extend the component-wise convergence to $\left (\mathbb{L}_4,d_{GHP}^4\right )$ . This is carried out exactly as in the proof of Lemma 3.3, following [Reference Addario-Berry, Broutin, Goldschmidt and Miermont4, Theorems 4.1, 4.4].