Proof. The first moment. Explore the cluster in a fixed mark-independent order, testing the mark of each newly discovered vertex immediately and stopping at the first mark or when the cluster is exhausted. No edge is queried twice. Critical finiteness from 2 makes this a terminating decision procedure, even in an infinite domain. Let \(T\) be its number of edge queries. Conditional on the complete percolation configuration, the discovery order is fixed. The first mark index is geometric with parameter \(h\), stopped at \(|C_x^D|\); conditional on success, it is that geometric variable conditioned to be at most \(|C_x^D|\). In both cases its mean is at most \(1/h\). There are at most \(d\) edge queries per discovered vertex, so \[\mathbb ET\le d/h,\qquad \mathbb E[T\mid E]\le d/h.\]

Let \(Z=\sum_{e\text{ queried}}(\omega_e-{p_c})\). The indicator \(Q_e\) that edge \(e\) is queried is measurable without its own bit, and every pivotal edge must be queried by a procedure deciding \(E\). Conditioning on all other edge bits and marks, and then averaging, gives \[\mathbb E[(\omega_e-{p_c})Q_e\mathbf 1_E] ={p_c}(1-{p_c})\mathbb P(e\text{ is pivotal}).\] The identity remains valid after summation, since \(\mathbb ET<\infty\). An open pivotal contributes to \(D_p\mathbf 1_E\), and its bit is independent of its pivotality. Therefore \[ \mathbb E[Z\mathbf 1_E]=(1-{p_c})\mathbb E[D_p\mathbf 1_E]. \tag{16}\]

Each queried bit is fresh conditional on the previous transcript. For a centered Bernoulli variable, the logarithmic moment-generating function is at most \(\lambda^2/8\): its second derivative is a Bernoulli variance, at most \(1/4\). Successive conditional expectations at the bounded stopping times \(T\wedge n\), followed by Fatou’s lemma, give \[\mathbb E\exp(\lambda Z-\lambda^2T/8)\le1.\] Put \(H=\mathbb P(E)\ge h\), since a mark at \(x\) suffices. Conditional Jensen’s inequality yields \[\lambda\mathbb E[Z\mid E] \le\log(1/H)+\frac{\lambda^2}{8}\mathbb E[T\mid E] \le\log(1/h)+\frac{\lambda^2d}{8h}.\] Taking \(\lambda=\sqrt{8h\log(1/h)/d}\) and using (16), we obtain \[\mathbb E[D_p\mid E] \le\frac{\sqrt{d/2}}{1-{p_c}} \sqrt{\frac{\log(1/h)}h}.\] This proves the first-moment assertion, uniformly over domains.

Higher binomial moments. On \(E\), each open pivotal is a bridge separating \(x\) from all marks in its cluster. These pivotals are linearly ordered from \(x\) toward the marks. Define \[b_i(D,x)=\mathbb E\left[\binom{D_p}{i}\mathbf 1_E\right], \qquad b_0(D,x)=\mathbb P(E).\] For an oriented edge \(\vec e=(a,b)\), let \(C_{x,\vec e}^D\) be the root cluster when \(e\) is forced closed. Decompose a selected set of \(i\) pivotals according to its first selected edge, oriented away from \(x\). The resulting exact identity is \[ b_i(D,x)={p_c}\sum_{\vec e=(a,b)\in E(D)} \sum_{\substack{W\subset D\text{ finite}\\x,a\in W, b\notin W}} \mathbb P_{p_c}(C_{x,\vec e}^D=W)(1-h)^{|W|} b_{i-1}(D\setminus W,b). \tag{17}\] To verify the conditional law in this formula, force \(e\) closed and expose the exact finite root cluster \(W\). This reveals internal and boundary information of \(W\) but no edges internal to \(D\setminus W\). The event \(C_{x,\vec e}^D=W\) ignores the actual bit of \(e\). Require that \(W\) contain no marks, accounting for \((1-h)^{|W|}\). Opening \(e\), with independent probability \({p_c}\), makes it the sole open contact from \(W\) to its exterior. It is then pivotal exactly when \(b\) connects to a mark in \(D\setminus W\). All pivotals after it are precisely those of that exterior event. The exterior edges and marks retain their independent laws. Every selected subset is counted once: if the total number of ordered pivotals is \(n\), the count is \(\sum_{j=1}^n\binom{n-j}{i-1}=\binom ni\). All sums are nonnegative, so Tonelli’s theorem justifies the decomposition without a priori moment assumptions.

Uniform induction in \(D,x\) now gives \[b_i(D,x)\le A_h^{i-1}b_1(D,x)\le A_h^ib_0(D,x).\] Finally expand \((1+(2A_h)^{-1})^{D_p}\) and sum these binomial moments to get \[\mathbb E\left[\left(1+\frac1{2A_h}\right)^{D_p}\mathbf 1_E\right] \le2\mathbb P(E).\] Since \(\log(1+(2A_h)^{-1})\ge(4A_h)^{-1}\) for \(A_h\ge1\), Markov’s inequality proves the second part of (15). ◻

Proof. On the event on the left, choose a witness vertex \(y\) by a fixed ordering. Independently mark vertices with probability \(h\). The conditional probability that \(y\) is the sole mark in \(C_x\) is at least \(h(1-h)^M\). In that case every bridge separating \(x\) from \(y\) is pivotal for the marked event. By (15), with \(R=KM^s/2\), \[\mathbb P_{p_c}(s_x\le M,\operatorname{rad}_{\mathrm{piv}}\ge R) \le\frac{2}{h(1-h)^M}\exp\left(-\frac{R}{4A_h}\right).\] Set \[X=\frac{K^2M^{2\epsilon}}{\log M},\qquad h=\frac{X}{bM} =\frac{K^2M^{-1+2\epsilon}}{b\log M},\] where \(b\) will be a sufficiently large fixed constant. The allowed range of \(K\) gives \(h\le1/(b\log M)\le1/2\). For all sufficiently large \(M\), uniformly in that range, \(\log(1/h)\le2\log M\) and \[\frac{R}{4A_h}\ge\frac{X}{8C\sqrt{2b}}.\] Using \(-\log(1-h)\le2h\), the logarithm of the preceding probability bound is therefore at most \[O(\log M)+\frac{2X}{b}-\frac{X}{8C\sqrt{2b}}.\] First choose \(b\) large enough that the last two terms have a negative sum bounded above by \(-c_1X\). Then increase \(M\): the minimum possible \(X\) is \(M^{2\epsilon}/\log M\), which dominates \(\log M\). This proves (18) with a fixed positive \(c\). ◻

Proof of 2. Given a capped cluster violating goodness, choose a deleted set \(A\) and a remaining component \(B\) witnessing the violation, by a fixed finite ordering. Let \(K\ge1\) be the number of original open edges from \(B\) to \(A\). Since \(\operatorname{diam}_{\mathrm{piv}}(B)\le|B|-1<M\), a violation has \(K\le M^{1-s}\).

Close its \(K\) open contacts, except for one retained contact if \(x\notin B\): in that case take a simple open path from \(x\) to \(B\) and retain its first edge entering \(B\). The path before that contact does not use any edge being closed, so the modified root cluster \(C'_x\) still contains \(B\). If \(x\in B\), then \(C'_x=B\). In either case \(B\) has at most one open attachment to the rest of \(C'_x\). An internal bridge of \(B\) cannot be bypassed outside \(B\), because such a bypass would need two attachments. Thus its bridge tree embeds into the bridge tree of \(C'_x\) with its distances preserved. The triangle inequality shows that at least one endpoint of a diameter of \(B\) is at pivotal distance greater than \(KM^s/2\) from \(x\). Also \(|C'_x|\le|C_x|\le M\). Hence the modified configuration belongs to the event in (18).

It remains to bound the cost of this configuration change; no independence of its input-dependent choices is needed. At a fixed output, every changed edge is incident to \(B\subseteq C'_x\), so there are at most \(dM\) candidate edges. For fixed \(K\), specifying the subset of at most \(K\) changed edges reconstructs the input by reopening them. The number of possible inputs is at most \[\sum_{j=0}^K\binom{\lfloor dM\rfloor}{j}\le(K+1)(dM)^K,\] and their probability ratio to the output is at most \(\max(1,{p_c}/(1-{p_c}))^K\). This is a comparison on a finite product: on the cap event every cluster vertex lies within distance \(M-1\) of \(x\), and every incident edge lies in a deterministic slightly larger ball. Condition on all bits outside that ball and apply the pointwise finite-product comparison inside it.

It follows from (18) that the contribution for fixed \(K\) is at most \[\exp\left(C_1K\log M- \frac{cK^2M^{2\epsilon}}{\log M}\right).\] Uniformly for \(K\ge1\), the negative term absorbs the positive one, because \(M^{2\epsilon}/(\log M)^2\to\infty\). Summing over the at most \(M\) possible values of \(K\) gives, for large \(M\), \[M\exp\left(-\frac{cM^{2\epsilon}}{2\log M}\right) \le e^{-M^\epsilon},\] as claimed. Finally, explore the root cluster until it is exhausted or more than \(M\) vertices are found. This uses only a deterministic ball of radius at most \(\lceil M\rceil+1\). In the first case its internal edges and closed boundary are known, and one can test every subset deletion of the finite cluster. In the second case the capped event fails. This proves locality. ◻

Proof. First let \(D\) be finite. Select a maximizer by a fixed tie rule. For each \(w\in D\), take a percolation configuration \(\omega\) and expose the exact vertex set \(B=C_w^D(\omega)\). Replace every edge with at least one endpoint in \(B\) by an independent Bernoulli(\(p\)) bit, leaving all other edges unchanged; call the resulting configuration \(\widetilde\omega\). Conditional on \(B\), the unchanged exterior edges have product law, and all replaced bits are fresh. Consequently \(\widetilde\omega\) has product law independent of \(B\). If \(Z\) is the selected maximizer in \(\mathcal B^D(v,M;\widetilde\omega)\), this independence gives \[\sum_{w\in D}\mathbb P(Z\in B) =\sum_{z\in D}\mathbb P(Z=z)\sum_{w\in D}T_p^D(w,z) =\mathbb E_p\max_{z\in\mathcal B^D(v,M)}f_D(z).\] Here a separate coupling is used for each \(w\); only the marginal law of \(Z\) needs to be the same in these couplings.

If \(v\notin B\) and \(Z\in B\), take the first edge entering \(B\) along a simple \(\widetilde\omega\)-open path from \(v\) to \(Z\) of length at most \(M\). This edge was closed in \(\omega\), and its exterior endpoint was reachable from \(v\) in \(D\setminus B\) by at most \(M-1\) open edges of \(\omega\). Let \(N_w(\omega)\) count boundary edges with this property, setting it to zero when \(v\in B\). Then \[\mathbb P(Z\in B)\le T_p^D(v,w)+\mathbb E_p N_w.\] To bound the last expectation, open one counted, distinguished edge. In the new configuration it is pivotal for \(v\leftrightarrow w\): it is the only open edge entering the old cluster \(B\). Moreover it is among the first \(M\) bridges separating \(v\) from \(w\), in their order from \(v\), because a simple path reaches it using at most \(M\) edges. For a fixed output and a distinguished edge, closing that edge recovers the input. There are at most \(M\) possible distinguished edges in each connected output. Comparing the two possible values of that one bit therefore gives \[\mathbb E_p N_w\le \frac{1-p}{p}M\,T_p^D(v,w).\] Summation over \(w\) proves (28). For an infinite domain, exhaust \(D\) by finite induced domains containing \(v\). Their susceptibilities and intrinsic balls increase to the corresponding objects in \(D\). The maximum also converges monotonically: the intrinsic \(M\)-ball is contained in a deterministic finite graph ball, and every witnessing path is finite. Monotone convergence proves the assertion. The case \(M=0\) is immediate. ◻

Proof. The unconditional endpoint kernel has finite range and continuous entries as a function of \(p\): its rule queries only finitely many edge bits, and its auxiliary randomization has a fixed law. Covariance makes its row sums constant on \(O\). The absolute difference of two such kernels is invariant on \(O\times O\), so (10) gives equal row and column sums. The finite range and the row–column norm bound therefore give operator-norm continuity. Dividing by the positive scalar \(u_p\to u\) proves the assertions after shortening the interval. The kernel \(K_p\) is stochastic and invariant, and hence has column sums one, although it need not be symmetric. The cap and connection conditions are properties of the accepted local patterns, so every success at \(p\) still has a simple open path of length at most \(M\) to its endpoint.

Choose such a path on each success. If it avoids \(A\), its endpoint lies in the intrinsic \(M\)-ball in \(V\setminus A\). Thus 20 bounds its unconditional contribution to the endpoint potential by \(\bigl(1+(1-p)M/p\bigr)F_A(v)\); this case is empty if \(v\in A\). Endpoints in \(A\) contribute zero. In the remaining case let \(a\sim b\) be the last exit of the chosen path from \(A\). Its terminal segment from \(b\) to the endpoint \(y\) lies in \(V\setminus A\) and has length \(l\le M\). Positive association with the event that this deterministic segment is open yields \[f_{V\setminus A}(b) \ge \mathbb E_p\bigl[\lvert C_y^{V\setminus A}\rvert; \text{the segment is open}\bigr] \ge p^l f_{V\setminus A}(y).\] Therefore \(F_A(y)\le p^{-M}F_A(b)\). Since \(\mathop{\mathrm{dist}}(v,a)\le M\), summing over all possible last exits bounds this contribution by \(p^{-M}h_A(v)\). Divide the two bounds by \(u_p\). Because \(p\ge{p_c}/2\) and \(u_p\ge u/2\), the first coefficient can be enlarged to a constant \(Z\in[2,CM/u]\), and the second to a finite constant depending on the fixed map. This proves (29). ◻

Proof. Assume (19), and define \[D(p)=\chi^{-2}\left[ \sum_{b\sim o}\mathbb E_p(s_os_b;\ C_o\ne C_b)\right].\] By (12), \((1-p)\chi'=\chi^2D(p)\). Our first goal is to prove, for every \(\gamma>0\), that \(D(p)\ge c_\gamma\chi^{-\gamma}\) when \(p\) is sufficiently close to \({p_c}\). We will compare two bounds on the exterior potential reached by repeated endpoint-map steps: (29) bounds it above in terms of \(D(p)\), while the small norm of \(K_p\) bounds the contribution of endpoints in the root’s own cluster and hence leaves a positive exterior contribution.

Fix a map and its interval as in 21. For \(x\in O\), expose \(A=C_x\) in an independent \(p\)-field, and use the possibly unnormalized expectation \[\widehat\mathbb E[G(A)]=\mathbb E_p\left[\frac{s_x}{\chi}G(C_x)\right].\] For each fixed vertex \(a\), exposure of the exact cluster \(C_a\) leaves fresh exterior edges and hence gives \[\begin{align*} \widehat\mathbb E\left[\mathbf 1_{\{a\in A\}} \sum_{b\sim a}F_A(b)\right] &=\chi^{-2}\sum_{b\sim a} \mathbb E_p\left[s_a\mathbf 1_{\{x\in C_a\}}f_{V\setminus C_a}(b)\right] \\ &\le \chi^{-2}\sum_{b\sim a} \mathbb E_p(s_as_b;\ C_a\ne C_b) \le c_\sigma D(p), \end{align*}\] where \(c_\sigma=(\min_i\sigma_i)^{-1}\). The last inequality follows because the preceding nonnegative expression is constant on each vertex type and its type average is \(D(p)\). Bounded degree now gives \(\widehat\mathbb Eh_A(v)\le C_{\mathrm{map}}D(p)\) uniformly in \(v\). Iterate (29), use stochasticity of \(K_p\), and note that \(F_A(x)=0\). Since \(Z\ge2\), the geometric sum gives \[ \widehat\mathbb E(K_p^jF_A)(x) \le C_{\mathrm{map}} Z^jD(p),\qquad j\ge1. \tag{31}\]

We next give a lower bound for the same quantity. Fresh exterior exposure identifies it as \[\widehat\mathbb E(K_p^jF_A)(x) =\chi^{-2}\sum_{v\in O}K_p^j(x,v) \mathbb E_p(s_xs_v;\ C_x\ne C_v).\] Without the distinctness condition, positive association and the uniform near-critical comparison of susceptibilities give a lower bound \(c_0>0\) for this normalized sum. To control what is removed, write \(I=C_x\cap O\), which is nonempty. Mass transport restricted to \(O\) gives the exact identity \[ \sum_{v\in O}K_p^j(x,v)\mathbb E_p(s_x^2;\ v\in C_x) =\mathbb E_p\left[\frac{s_x^2}{\lvert I\rvert} \langle\mathbf 1_I,K_p^j\mathbf 1_I\rangle\right] \le \lVert K_p\rVert^{j}\mathbb E_p s_x^2 \le C\lVert K_p\rVert^{j}\chi^3. \tag{32}\] For completeness, the transport sends, from \(z\in O\) to each \(y\in C_z\cap O\), the mass \[\frac{s_z^2}{\lvert C_z\cap O\rvert} \sum_{v\in C_z\cap O}K_p^j(z,v).\] Its outgoing and incoming expectations are the two sides of the identity. The norm bound uses \(\lVert\mathbf 1_I\rVert_2^2=\lvert I\rvert\); it does not require symmetry of \(K_p\). Finally, the three-arm estimate (9) gives \(\mathbb E_p s_x^2\le\chi_{\max}^3\le C\chi^3\).

Choose \(C_*>1\) large enough that \[j=\left\lceil\frac{\log(C_*\chi)}{H-1}\right\rceil\] makes the normalized loss in (32) at most \(c_0/2\). Combining this lower bound with (31) and \(\log Z\le U+m+C'\) gives \[ D(p)\ge c_{\mathrm{map}} \chi^{-(U+m+C')/(H-1)}. \tag{33}\] Here \(C'\) is graph-dependent; the prefactor may depend on the fixed map. Since \(U+m\ge\log2\), the maps with \((U+m)/H\to0\) have \(H\to\infty\). Thus, for any \(0<\gamma<1\), we can fix a map for which the exponent in (33) is at most \(\gamma\). The susceptibility identity (12) then gives, almost everywhere on its interval, \(\chi'\ge c\chi^{2-\gamma}\). In particular \[\frac{d}{dp}\chi^{-(1-\gamma)} \le -(1-\gamma)c.\] Integrating from \(p\) to \(r<{p_c}\), discarding the nonnegative value at \(r\), and then letting \(r\uparrow{p_c}\) yields \[ \chi(p)\le C_\gamma({p_c}-p)^{-1/(1-\gamma)}. \tag{34}\] Local absolute continuity, established with (12), justifies this differential calculation.

We transfer this bound to a critical tail by an explicit comparison of exploration transcripts. For a large integer \(N\), put \(p={p_c}-N^{-1/2}\). From any vertex \(v\), explore until the cluster is exhausted or \(N\) vertices have been discovered. Use a fixed nonretesting exploration order. There are at most \(dN\) fresh edge queries, and the transcript decides the event \(s_v\ge N\). If \(L_N\) is the likelihood ratio of its law at \({p_c}\) relative to its law at \(p\), each active query contributes either \({p_c}/p\) or \((1-{p_c})/(1-p)\). For fixed \(b>1\), its conditional \(b\)th-moment factor under \(p\) is \[p({p_c}/p)^b+(1-p)((1-{p_c})/(1-p))^b \le 1+C_b({p_c}-p)^2.\] Indeed this expression is twice continuously differentiable in a fixed neighborhood of the interior point \({p_c}\), equals one there, and has first derivative zero there. Pad the stopped exploration with inactive factors equal to one. Successive conditioning for the deterministic \(dN\) steps gives \[\mathbb E_p L_N^b\le(1+C_b/N)^{dN}\le e^{dC_b}.\] Hölder’s inequality and Markov’s inequality therefore imply, uniformly in \(v\), \[ \mathbb P_{{p_c}}(s_v\ge N) =\mathbb E_p[L_N\mathbf 1_{\{s_v\ge N\}}] \le C_b'\left(\frac{\chi_v(p)}{N}\right)^{1-1/b}. \tag{35}\] Choose \(\gamma\) small and then \(b\) large enough that \[\theta=(1-1/b)\left(1-\frac1{2(1-\gamma)}\right)>\alpha,\] where \(\alpha\) is the exponent in (19). For all sufficiently large \(N\), (34) applies to the chosen \(p\); the uniform type comparison and (35) give \(\mathbb P_{{p_c}}(s_v\ge N)\le C N^{-\theta}\). Summing \(((N+1)^\alpha-N^\alpha)\mathbb P_{{p_c}}(s_v>N)\) proves a uniform finite \(\alpha\)th moment, contrary to (19). This establishes (30) for every \(0<\alpha<1/2\). ◻

Proof. Take an independent product copy \(Y\). Run the algorithm on \(X\), and let \(Z^t\) replace the first \(t\) queried coordinates of \(X\) by their values in \(Y\). Include the algorithm’s independent seed in each transcript. Conditional on any pre-query transcript, the unqueried coordinates of \(X\) retain their independent original marginals, since that transcript constrains only queried bits. The replaced coordinates of \(Y\) are independent of this transcript and of those unqueried bits. Thus the pre-query hybrid has the original product law conditional on the transcript. If the next index is \(i\), that index is transcript-measurable and \(Y_i\) is still fresh. Consequently \[\mathbb E\bigl[\lvert f(Z^{t-1})-f(Z^t)\rvert \mid\text{pre-query transcript}\bigr] =\mathbb E\lvert f(X)-f(X^{(i)})\rvert.\] At the terminal transcript, the same reasoning gives product law for \(Z^T\) conditional on the entire transcript. Since the transcript determines \(g(X)\), it follows that \[\mathop{\mathrm{Cov}}(f(X),g(X))=\mathbb E[(f(X)-f(Z^T))g(X)].\] Telescope and use \(0\le g\le1\) before conditioning on partial transcripts. The preceding conditional-increment identity then proves (36), because each coordinate is queried at most once. Intermediate hybrids need not be independent of \(g(X)\). ◻

Proof. Fix \(0<a<1/2\). In a finite induced domain, fix a vertex \(v\) and an integer \(n\ge2\). Let \(f\) indicate \(\lvert C_v\rvert\ge n\). Independently mark every vertex with probability \(1/n\), and let \(g\) indicate that \(C_v\) contains a mark. An algorithm deciding \(g\) first reveals all marks and then explores the clusters of marked vertices. It queries an edge only if at least one endpoint belongs to a cluster containing a mark. By 4, restriction coupling, and monotonicity in \(p\le{p_c}\), each vertex has this property with probability at most \[\mathbb E_p[1-(1-1/n)^{s_v}] \le\mathbb E_p\min(1,s_v/n) \le n^{-a}\mathbb E_p s_v^a\le C_a n^{-a}.\] Thus every edge has revealment at most \(2C_a n^{-a}\), uniformly in the domain, the root, and \(p\le{p_c}\). Mark coordinates have zero influence on \(f\). For an increasing Boolean edge function, its absolute resampling influence is exactly \(2p(1-p)\) times the probability that the edge is pivotal. Writing \(F(p)=\mathbb P_p(s_v\ge n)\), finite-product differentiation and 22 give \[\mathop{\mathrm{Cov}}(f,g)\le C_a n^{-a}\,2p(1-p)F'(p).\] On the other hand, conditioning on the percolation configuration gives \[\mathop{\mathrm{Cov}}(f,g) \ge\bigl(1-(1-1/n)^n-C_a n^{-a}\bigr)F(p).\] For all sufficiently large \(n\), the factor in parentheses is bounded below by a positive constant. Since \(2p(1-p)\le1/2\), we obtain \(F'(p)\ge c_a n^a F(p)\) for \(0<p\le{p_c}\). Multiplying by \(\exp(-c_an^ap)\) shows that \(\exp(-c_an^ap)F(p)\) is nondecreasing. Integrating up to \({p_c}\) therefore gives, including the case \(F\equiv0\), \[ \mathbb P_p(s_v\ge n) \le C_a n^{-a}\exp\{-c_a({p_c}-p)n^a\}. \tag{38}\] We used the critical moment bound to estimate \(F({p_c})\). Continuity gives the assertion also at \(p=0\). Increasing finite domain exhaustion is legitimate because the event \(s_v\ge n\) has a finite open witness; all constants are independent of the domain. Enlarge the constant to cover the finitely many smaller values of \(n\).

Put \(\delta={p_c}-p\). The function \(x^{-a}e^{-c_a\delta x^a}\) is decreasing on \((0,\infty)\). The substitution \(t=c_a\delta x^a\) in its integral gives \[\sum_{n\ge1}n^{-a}e^{-c_a\delta n^a} \le C_a\delta^{-(1-a)/a}.\] Summing (38) therefore proves the same bound for \(\chi_{\max}\). Finally take \(a=1/(2+\eta)\), for which \((1-a)/a=1+\eta\), to obtain (37). ◻

Proof. Fix an oriented graph edge \((x,v)\) and \(n\ge1\). Let \[H_n=\{C_x\ne C_v,\ n\le\min(s_x,s_v)<2n\}.\] On distinct clusters the two size events have disjoint finite open witnesses. The BK inequality, coupling with critical percolation, and (30) give \[ \mathbb P_p(H_n)\le C n^{-2\alpha}. \tag{41}\] We also need to control their number of contacts, and for this we compare two exploration scores.

First explore \(C_v\) in the full graph, querying every edge incident to each discovered vertex exactly once, including edges whose endpoints have both already been discovered. Stop after \(N=\lceil2dn\rceil\) queries if the exploration has not finished. Let \(S_{\mathrm{full}}\) be the sum of \(\omega_f-p\) over the tested edges. Every active query is a fresh Bernoulli(\(p\)) bit. Padding after stopping by zero increments, the elementary centered Bernoulli exponential bound gives \[\mathbb E_p e^{\lambda S_{\mathrm{full}}}\le e^{N\lambda^2/8}, \qquad \mathbb P_p(\lvert S_{\mathrm{full}}\rvert>t) \le2e^{-2t^2/N}.\] This is an unconditional estimate, with no cluster-size event imposed.

On the same percolation configuration, perform a second exploration by first exposing \(A=C_x\) together with its internal bits and closed boundary. If \(v\notin A\), explore the cluster of \(v\) in \(V\setminus A\), again querying every incident edge of that domain and using the same deterministic query cap \(N\). Let the resulting score be \(S_{\mathrm{ext}}\), and set it to zero if \(v\in A\). Conditional on the exposure, all the queried exterior bits are fresh and independent. The identical tail bound therefore holds conditionally, uniformly in \(A\), and hence also unconditionally after averaging over \(A\).

On \(H_n\cap\{s_v\le s_x\}\), both explorations finish the same cluster \(B=C_v\), because \(\lvert B\rvert<2n\) and its number of incident edges is at most \(d\lvert B\rvert<N\). The completed queried sets are, respectively, all full-graph edges incident to \(B\) and all such edges with both ends outside \(A\). Their difference consists exactly of the \(A\)–\(B\) contacts, which are closed. Query order is irrelevant to these completed sums, so \[S_{\mathrm{full}}-S_{\mathrm{ext}}=-p k_e(A,B).\] It follows that the event under consideration with \(k_e(A,B)>2t/p\) is contained in the union of the two score-tail events. Take \(t=n^{1/2+\epsilon_1}\), and use \(p\ge{p_c}/2\). Interchanging \(x\) and \(v\) handles the opposite size order. Thus \[\mathbb P_p\left(H_n\cap \{k_e(C_x,C_v)>C n^{1/2+\epsilon_1}\}\right) \le C e^{-c n^{2\epsilon_1}}.\] Each score was concentrated in its own fresh law before intersecting with \(H_n\); no independence between the scores is asserted. Since \(k_e\le d\min(s_x,s_v)<2dn\) on \(H_n\), (41) now yields \[\mathbb E_p[k_e(C_x,C_v)\mathbf 1_{H_n}] \le C n^{1/2+\epsilon_1-2\alpha} +C n e^{-c n^{2\epsilon_1}} \le C n^r.\] Apply (39) with one contact factor and partition \(\min(\lvert A\rvert,\lvert B\rvert)\ge L\) into the bins \([2^jL,2^{j+1}L)\), \(j\ge0\). There are at most \(d\) neighbors of each root, and \(\sum_{j\ge0}(2^jL)^r\le C_rL^r\) because \(r<0\). This proves (40). ◻

Proof. All operators below are bounded at each subcritical parameter. The operator-norm local absolute continuity supplied by (12), and cyclicity of the finite trace, give \(a_3'=3\mathop{\mathrm{tr}}(T^2T')\) almost everywhere. Set \(J=T^2\). For fixed terminals \(x,y\), the first identity in (12) can also be written \[(1-p)T'_p(x,y) =\mathbb E_p[k_e(C_x,C_y)\mathbf 1_{\{C_x\ne C_y\}}].\] Each edge between the two terminal clusters has exactly one orientation from the first cluster to the second. Applying (39) to \(J\) therefore gives \[ (1-p)a_3'=3\mathcal D(k_e k_J) \quad\text{for almost every }p<{p_c}. \tag{43}\]

We will use the following square bound: \[ \mathcal D(k_J^2)\le\mathop{\mathrm{tr}}(JTJT)=a_6\le m_p^3a_3. \tag{44}\] To check the first inequality, reroot one \(J\) factor using (39) and expand the other one. This gives \[\left[\sum_{v,x',y'}J(o,v)J(x',y') \mathbb P_p(o\leftrightarrow x',\ v\leftrightarrow y',\ C_o\ne C_v)\right].\] The two connections on this event have disjoint open witnesses, so BK bounds their probability by \(T(o,x')T(v,y')\). Symmetry of \(J\) and \(T\) identifies the resulting sum with \(\mathop{\mathrm{tr}}(JTJT)\). Because \(J=T^2\), this is \(a_6\). Finally, positivity of \(T\) gives \(T^6\le m_p^3T^3\) in quadratic-form order, which proves the remaining inequality after taking the trace.

Choose \(\alpha\) and \(\epsilon_1\) as in 23. Split (43) according to whether the smaller terminal cluster has size less than \(L\). For the part where \(A\) is small, reroot one contact and expose \(A=C_o\) with its internal bits and boundary. If \(v\notin A\), its exterior cluster is fresh, so \[\mathbb E_p[k_J(A,C_v)\mid A\text{ and its exposure}] \le\sum_{y\in A}\sum_z J(y,z)T(v,z) =\sum_{y\in A}(T^3)(y,v).\] The positive operator \(T^3\) satisfies \[\lvert(T^3)(y,v)\rvert \le\sqrt{(T^3)(y,y)(T^3)(v,v)}\le c_\sigma a_3,\] since every diagonal entry is bounded by \(a_3/\min_i\sigma_i\). Consequently \[\mathcal D(k_e k_J\mathbf 1_{\{\lvert A\rvert<L\}}) \le d c_\sigma a_3\,[\mathbb E_p(s_o\mathbf 1_{\{s_o<L\}})] \le C L^{1-\alpha}a_3.\] Here \(s\mathbf 1_{\{s<L\}}\le L^{1-\alpha}s^\alpha\) and (30) supply the last bound. Reversing the ordered cluster pair gives the same estimate when \(B\) is small, since \(k_e\) and \(k_J\) are symmetric in their two arguments.

For the remaining part, Cauchy–Schwarz for the positive measure underlying \(\mathcal D\), followed by (40) and (44), gives \[\mathcal D(k_e k_J\mathbf 1_{\{\min(\lvert A\rvert,\lvert B\rvert)\ge L\}}) \le C L^{1/4+\epsilon_1/2-\alpha}m_p^{3/2}\sqrt{a_3}.\] The factor \((1-p)^{-1}\) in (43) is uniformly bounded on \([{p_c}/2,{p_c})\). Also \(a_3\ge1\), directly from the nonnegative kernel sum and \(T(o,o)=1\). Thus \(b=\sqrt{a_3}\ge1\) is locally absolutely continuous and satisfies \[ b'\le C L^{1-\alpha}b +C L^{1/4+\epsilon_1/2-\alpha}m_p^{3/2} \quad\text{almost everywhere}. \tag{45}\]

We make the exponent choices explicit. Given \(\eta>0\), choose \(0<\zeta<\min(1,\eta)\), and then choose \(1/2-\alpha>0\), \(\epsilon_1>0\), and \(\eta'>0\) sufficiently small that \(1/2+\epsilon_1-2\alpha<0\) and \[\begin{align*} A_*&=(2-\zeta)(1-\alpha)<1,\\ B_*&=(2-\zeta)(1/4+\epsilon_1/2-\alpha) +\tfrac32(1+\eta')<1+\eta/2. \end{align*}\] These choices are possible by continuity: at the limiting values \(\alpha=1/2\), \(\epsilon_1=\eta'=0\), the two expressions are \(1-\zeta/2\) and \(1+\zeta/4\), respectively. Put \(\delta={p_c}-p\) and \(L=\delta^{-(2-\zeta)}\ge1\). The susceptibility bound (37) implies \(m_p\le\chi_{\max}(p)\le C\delta^{-1-\eta'}\), so (45) becomes \[b'(p)\le C\delta^{-A_*}b(p)+C\delta^{-B_*}.\] Since \(A_*<1\), the integral of the first coefficient from \({p_c}/2\) to \({p_c}\) is finite. Multiplication by its integrating factor therefore gives \[b(p)\le C\left(b({p_c}/2)+ \int_{{p_c}/2}^p({p_c}-r)^{-B_*}\,dr\right) \le C_\eta\delta^{-\eta/2}.\] For the last inequality one may bound the integrand by a constant times \(({p_c}-r)^{-1-\eta/2}\), using \(B_*<1+\eta/2\). Squaring proves (42). ◻

Proof. Since \(\ell_w\le k\) and \(b_w\le dk\), there is a fixed \(B\ge1\) with \(p^{-\ell_w}q^{-b_w}\le B^k\). For a specified edge \(e\), symmetry and stochasticity of \(P\) show that the total probability, summed over all bases, of traversing \(e\) at a specified step is \(1/d\). A union bound over the \(k\) steps therefore gives \[\sum_{w:\,e\in E(\gamma_w)}c(w) \le B^k\sum_{u}\sum_{w:\,w\text{ traverses }e}\mu_u(w) \le B^k k/d.\] Increasing \(C\) proves (46) for every \(k\ge3\). Every counted template has all its path edges on every simple open \(x\)–\(y\) path. Charge it to an edge of one such path, which has fewer than \(|C_x^D|\) edges, to obtain the asserted bound on \(N^D_{xy}\).

To prove monotonicity, consider a pair already connected before an edge opening or a domain enlargement. Every edge of a template counted afterward belongs to every old simple connection. Its retained path was therefore already eligible, open, and pivotal. Each blocker closed or absent afterward was closed or absent before. Thus every template counted afterward was counted before, with exactly the same weight. Its total penalty cannot increase. A pair previously disconnected has weight zero, so monotonicity holds for all pairs.

Symmetry in the terminals follows directly from the pivotal condition; it does not require any reversal property of loop erasure. Invariance and entrywise domination are immediate, and the row and column sum bound for operator norms gives (47). Finally, \[\sum_y N^D_{xy}W_t^D(x,y)\le c_k|C_x^D|^2.\] The right side has finite expectation by (9) and subcritical susceptibility finiteness. Dominated convergence proves the differentiation and continuity assertions. These arguments apply to infinite domains: their clusters are almost surely finite, and the preceding bound is uniform under finite exhaustion. In fact, once an exhaustion of \(D\) contains the entire cluster \(C_x^D\), all weighted connections inside it agree with their limiting values: boundary blockers are closed both before and after their other endpoints enter the domain. ◻

Proof. All bits specified by \(Q_w\) are incident to \(I_w\), and there are exactly \(\ell_w\) specified open edges and \(b_w\) specified closed edges. This proves independence and (48). On \(Q_w\) each internal vertex has exactly the two corridor edges open. The resulting geometry is shown in 3. A simple connection using every corridor edge must traverse the whole corridor consecutively, with both terminals outside its interior. If the two exterior endpoint clusters coincide, there is an exterior bypass, so the corridor edges cannot all be pivotal. If the exterior clusters are distinct, the corridor is their only open link, so all of its edges are pivotal for every terminal pair in either displayed orientation. ◻

Proof. Expand \(-\partial_tT_t(x,y)\) as a nonnegative sum over templates and discard the factor \(e^{-tN^V_{xy}}\le1\). By 25, the term for \(w\) is at most \[\mu_u(w)\bigl(T_0(x,u)T_0(v,y)+T_0(x,v)T_0(u,y)\bigr).\] Indeed the two exterior connections in each orientation belong to distinct clusters, so BK bounds their joint probability by the product of the corresponding full-graph connection probabilities. Add the discarded raw paths to this upper bound and sum over bases and endpoints. Each orientation gives \(T_0P^kT_0\), since \(P^k\) is symmetric. Integration from zero to \(t\) proves the entrywise assertion. Domination by a nonnegative kernel gives \[\lVert T_0-T_t\rVert\le 2t\lVert T_0P^kT_0\rVert \le2t\rho^km_0^2.\] The triangle inequality proves the norm assertion. In particular, no positive-semidefiniteness assertion for \(T_t\) is used. ◻

Proof. We first compare weighted connections in a fixed configuration. Suppose \(W_t^D(v,y)>W_t^{D\setminus A}(v,y)\). There was an old connection. If deletion destroys it, an old simple \(v\)–\(y\) path visits \(A\); this includes the case of a deleted terminal. Otherwise some positive-weight template is newly counted after deletion. Its retained path already lay in \(D\) and was open there. The raw path and its weight have not changed, so there are only two possible reasons it was not counted before.

First, a path edge may previously have been nonpivotal. An old simple connection avoiding that edge must visit \(A\), since otherwise the edge would remain nonpivotal after deletion. Second, a blocker may previously have been open. Since it is now absent, it joins an internal retained-path vertex \(z\notin A\) to a vertex of \(A\). Every surviving simple connection uses the new template path and therefore visits this \(z\in\mathop{\mathrm{Nb}}[A]\). This second case includes a removed pendant branch: its attachment vertex, rather than the removed vertex, lies on the connection.

The entire positive loss, which is at most one, is thus bounded by the indicator that some old simple \(v\)–\(y\) path visits \(\mathop{\mathrm{Nb}}[A]\). Splitting such a path at a visited vertex \(z\) gives edge-disjoint connection witnesses from \(v\) to \(z\) and from \(z\) to \(y\). A union bound and BK yield \[\mathbb E_p\bigl(W_t^D(v,y)-W_t^{D\setminus A}(v,y)\bigr) \le\sum_{z\in\mathop{\mathrm{Nb}}[A]}T_0(v,z)T_0(z,y).\] There is no sum over penalty weights in this estimate: a single visiting event bounds the whole loss. Summing over \(y\) proves (51). For infinite \(A\) all sums are interpreted monotonically, with an infinite right-hand error giving a trivial bound. ◻

Proof. Expose the exact finite cluster \(A=C_u^D\), including every internal edge bit and its closed boundary. This determines \(X_t^D(u)\): every counted template has retained path in \(A\), its weight is deterministic, and its blockers leaving \(A\) are closed or absent. The edges with both endpoints in \(D\setminus A\) remain independent Bernoulli variables of parameter \(p\).

For terminals in a cluster outside \(A\), weighted connections before and after deleting \(A\) agree exactly. All open paths stay in that exterior cluster; every eligible counted retained path stays there too; the only blockers removed were closed contacts with \(A\). Raw excursions into \(A\) affect neither eligibility nor weight. Consequently, including \(v\in A\) by the zero convention, \[ \mathbb E_p\bigl(X_t^D(u)X_t^D(v);\ C_u^D\ne C_v^D\bigr) =\mathbb E_p\bigl[X_t^D(u)s_t^{D\setminus A}(v)\bigr]. \tag{53}\] Apply 26 inside this expectation and use \(X_t^D(u)\le |A|\). For adjacent \(a,z\), positive association with the event that their edge is open gives \(pT_0(v,z)\le T_0(v,a)\). Hence \[\sum_{z\in\mathop{\mathrm{Nb}}[A]}T_0(v,z) \le(1+d/p)\sum_{a\in A}T_0(v,a).\] The three-arm estimate (9), summed over a third terminal, gives \[\begin{split} \mathbb E_p\bigl(|A|\mathbf 1_{\{a\in A\}}\bigr) &=\sum_y\mathbb P_p(u\leftrightarrow a,\ u\leftrightarrow y\text{ in }D)\\ &\le\sum_b T_0(u,b)T_0(b,a)\sum_y T_0(b,y) \le\chi_0^+T_0^2(u,a). \end{split}\] The error in (53) is therefore at most \[(1+d/p)(\chi_0^+)^2\sum_a T_0^2(u,a)T_0(a,v),\] which proves (52) with \(C'=1+2d/{p_c}\). ◻

Proof. The total weight of all templates whose retained paths use any edge of \(\gamma_w\) is at most \(\ell_wc_k/k\le c_k\), by (46). Consider any remaining template counted for \(x\leftrightarrow y\). Its retained path cannot meet \(I_w\): every open edge incident to \(I_w\) is an edge of \(\gamma_w\). Its connected retained path thus lies in one of the two distinct exterior endpoint clusters.

If it lies in the \(u\) cluster, every one of its edges is pivotal for the exterior connection \(x\leftrightarrow u\). Otherwise an exterior \(x\)–\(u\) path avoiding that edge, followed by the corridor and an exterior \(v\)–\(y\) path, would give a global \(x\)–\(y\) connection avoiding it. Its blockers remain closed or absent on restriction to \(D_w\). Thus it is counted, with the same weight, in \(N^{D_w}_{xu}\). The \(v\) cluster gives \(N^{D_w}_{vy}\) in the same way. This proves the bound. ◻

Proof of 8. Expand the negative derivative of \(\chi_t\) as the sum over the starting terminal, the other terminal, and all counted templates. Use the mass-transport identity (10) to move the root from the starting terminal to the template base. All terms are nonnegative, so this rearrangement is justified by Tonelli’s theorem. For each template retain only the orientation \(x\in C_u^{D_w}\), \(y\in C_v^{D_w}\). On \(Q_w\) and this exterior event, 28 implies \[W_t^V(x,y)\ge e^{-tc_k}W_t^{D_w}(x,u)W_t^{D_w}(v,y).\] Independence of \(Q_w\) from the exterior and (48) replace its factor \(c(w)\) by \(\mu_u(w)\). After summing over \(x,y\), we obtain \[-\chi_t'\ge e^{-tc_k}\mathbb E_{\rm path}\left[ \mathbf 1_{\{\ell_w\ge3\}} \mathbb E_p\bigl(X_t^{D_w}(u)X_t^{D_w}(v); C_u^{D_w}\ne C_v^{D_w}\bigr)\right].\] Apply 27. For its nonnegative error kernel, adding discarded paths only increases the average, and \[\mathbb E_{\rm path}T_0^3(u,v) =[\sum_v P^k(o,v)T_0^3(v,o)] =\mathop{\mathrm{tr}}(P^kT_0^3)\le\rho^k\mathop{\mathrm{tr}}(T_0^3).\] The last inequality uses the positive operator \(T_0^3\) and the positive trace inequality of 6 for the self-adjoint operator \(P^k\). It requires no commutation of \(P\) with \(T_0\). This proves (50). ◻

Proof. Let \(F\) be the event that this fixed path is open. On \(F\) and on a connection from \(z\) to \(y\), there is also a connection from \(b\) to \(y\). Templates counted for the latter connection whose retained paths use an edge of the fixed path have total weight at most \(lc_k\), by (46). For any other counted template, every retained edge is pivotal for \(z\leftrightarrow y\): a \(z\)–\(y\) bypass, concatenated with the fixed \(b\)–\(z\) path, would otherwise give a \(b\)–\(y\) bypass. Its blocker condition is unchanged. Thus, on \(F\), \[W_t^D(b,y)\ge e^{-tlc_k}W_t^D(z,y).\] Both \(\mathbf 1_F\) and \(W_t^D(z,y)\) are increasing functions of the percolation bits, by 24. Positive association and \(\mathbb P_p(F)=p^l\) give \[\mathbb E_pW_t^D(b,y) \ge e^{-tlc_k}\mathbb E_p[\mathbf 1_FW_t^D(z,y)] \ge e^{-tlc_k}p^l\mathbb E_pW_t^D(z,y).\] Summation over \(y\) proves (55). ◻

Proof. Positive-semidefinite Cauchy–Schwarz for \(T_0\) gives \[\lvert(P^lT_0)(a,b)\rvert^2 \le (P^lT_0P^l)(a,a)T_0(b,b).\] Here \(T_0(b,b)=1\). The invariant positive operator \(P^lT_0P^l\) has every diagonal entry bounded by \(c_\sigma\) times its weighted trace. Consequently, trace cyclicity and the positive-trace bound give \[(P^lT_0P^l)(a,a) \le c_\sigma\mathop{\mathrm{tr}}(P^{2l}T_0) \le c_\sigma\lVert P^{2l}\rVert\mathop{\mathrm{tr}}(T_0) =c_\sigma\rho^{2l}.\] This argument uses no commutation between \(P\) and \(T_0\). ◻

Proof of 30. Let \(L_{\mathrm{type}}\) and \(c_*>0\) be such that every transition probability of the finite type chain at every time \(L\ge L_{\mathrm{type}}\) is at least \(c_*\). The same \(c_*\) works for all types and will therefore remain fixed as \(J\) varies.

Proof of 2. Suppose that the critical connection kernel \(T_{{p_c}}\) is unbounded. Put \(p_0={p_c}/2\). For \(p_0\le p<{p_c}\), use the corridor notation \[\delta={p_c}-p,\qquad S=\log(1/\delta),\qquad m_0=\lVert T_p\rVert,\qquad R=\frac{\chi_0^+}{m_0}\ge1, \qquad a_3=\mathop{\mathrm{tr}}(T_p^3).\] Sprinkling (13) with \(a=\delta/(1-p)\) would bound \(T_{{p_c}}\) if \(ad m_0<1\). The assumed unboundedness, together with (37) and (42), therefore gives \[ m_0\ge\frac{1-p}{d\delta},\qquad \log R=o(S),\qquad \log a_3=o(S) \quad(\delta\downarrow0). \tag{65}\] Indeed \(m_0\ge c\delta^{-1}\), whereas, for every \(\eta>0\), \(\chi_0^+\le C_\eta\delta^{-1-\eta}\) and \(1\le a_3\le C_\eta\delta^{-\eta}\). Hence \(1\le R\le C'_\eta\delta^{-\eta}\) for every \(\eta>0\).

Fix \(0<\tau<c_f/(4c_\sigma)\) and the constants of 30 for this value of \(\tau\). They are now fixed throughout the limit \(p\uparrow{p_c}\). Since \(j\le J+1+C_1\log((k+1)R)\), there are fixed constants \(c>0\) and \(C_2>0\) such that \[ p^{2j}\ge c(kR)^{-C_2}\qquad(k\ge k_0,\ R\ge1). \tag{67}\] For example, one may take \(C_2=2C_1\log(1/p_0)\) and \(c=p_0^{2(J+1)}2^{-C_2}\).

Put \(\lambda=-\log\rho>0\) and choose a fixed \(K_*\) with \(\kappa:=\lambda K_*>C_2+4\). Set \[A_\delta=\log(2R)+\log(2+a_3)+\log S,\qquad k=\lceil K_*A_\delta\rceil,\qquad B_*=(kR)^{C_2+2},\qquad t_*=\frac{B_*}{m_0}.\] By (65), \(k\to\infty\) and \(k=o(S)\); in particular \(k\ge k_0\) and \(k\le S\) eventually. The estimates needed below are \[ B_*\rho^k=o(1),\qquad R^2a_3\rho^k=o((kR)^{-C_2}),\qquad t_*jc_k=o(1). \tag{68}\] To check their simultaneous validity, the definition of \(k\) gives \[\rho^k\le (2R)^{-\kappa}(2+a_3)^{-\kappa}S^{-\kappa}.\] Since \(R,a_3\ge1\) and \(k\le S\), it follows that \[B_*\rho^k\le C S^{C_2+2-\kappa}\longrightarrow0, \qquad (kR)^{C_2}R^2a_3\rho^k \le C S^{C_2-\kappa}\longrightarrow0.\] Finally \(c_k=e^{Ck}\) and \(m_0\ge c e^S\), so \[\log(t_*jc_k) \le -S+(C_2+2)(\log k+\log R)+\log j+Ck+O(1) =-S+o(S).\] This proves (68) with every path constant already fixed.

Consider the deformation with these values of \(p,k\), on \(0\le t\le t_*\). Equation (49) and the first estimate in (68) give \[ m_t\ge m_0(1-2B_*\rho^k)\ge\frac{m_0}{2},\qquad \chi_t\ge\frac{m_0}{2c_\sigma}. \tag{69}\] The second inequality uses the row-sum bound \(m_t\le c_\sigma\chi_t\); it requires no positivity of \(T_t\) as an operator.

For a selected raw path in 30, let \(z\) be either of its two escape endpoints. By (51), (56), (66), and (69), \[\begin{split} s_t^{D_w}(z) &\ge c_f\chi_t-\chi_0^+\frac{\tau}{R} =c_f\chi_t-\tau m_0\\ &\ge(c_f-2c_\sigma\tau)\chi_t \ge\frac{c_f}{2}\chi_t. \end{split}\] The hypotheses of (66) hold throughout this interval by the last estimate in (68). Apply (55) to the two simple escape paths in \(D_w\), each of length at most \(j\). Their product is bounded below by \[s_t^{D_w}(u)s_t^{D_w}(v) \ge e^{-2tjc_k}p^{2j}\left(\frac{c_f}{2}\chi_t\right)^2.\] Such paths have \(\mathbb E_{\mathrm{path}}\)-probability at least \(b_*\), so (67) and (68) imply \[ \mathbb E_{\mathrm{path}}\!\left[ \mathbf 1_{\{\ell_w\ge3\}}s_t^{D_w}(u)s_t^{D_w}(v)\right] \ge c_1(kR)^{-C_2}\chi_t^2, \tag{70}\] uniformly on \([0,t_*]\), with \(c_1>0\) independent of sufficiently small \(\delta\). On the other hand, the error in (50) is at most \[C'(\chi_0^+)^2\rho^ka_3 =C'R^2m_0^2\rho^ka_3 \le4C'c_\sigma^2R^2\rho^ka_3\chi_t^2 =o((kR)^{-C_2}\chi_t^2).\] The prefactor \(e^{-tc_k}\) in that equation is uniformly bounded below. We conclude that, for a constant \(c_2>0\), \[ -\chi_t'\ge c_2(kR)^{-C_2}\chi_t^2 \qquad(0\le t\le t_*). \tag{71}\] The differentiability and domination established for the deformation justify integration. Since \(\chi_t>0\), it yields \[\frac1{\chi_{t_*}} \ge c_2(kR)^{-C_2}t_* =\frac{c_2(kR)^2}{m_0}.\] But (69) gives \(1/\chi_{t_*}\le2c_\sigma/m_0\). These inequalities are incompatible with \(kR\to\infty\). Thus \(T_{{p_c}}\) is bounded, proving 2. ◻

Proof. Each finite-domain connection probability is continuous in \(p\), and connectivity in \(G\) is the increasing union of finite-domain connectivity events. Thus \(T_p(x,y)\uparrow T_{p_*}(x,y)\) as \(p\uparrow p_*\). If the subcritical norms are bounded by \(M\), then for finitely supported \(f,g\) the finite sum \(\langle f,T_pg\rangle\) tends to \(\langle f,T_{p_*}g\rangle\) and has absolute value at most \(M\lVert f\rVert_2\lVert g\rVert_2\). The limiting sesquilinear form extends by density to a bounded operator with the required matrix entries and norm at most \(M\). Conversely, entrywise nonnegativity gives \[|\langle f,T_pg\rangle|\le\langle|f|,T_p|g|\rangle \le\langle|f|,T_{p_*}|g|\rangle \le\lVert T_{p_*}\rVert\lVert f\rVert_2\lVert g\rVert_2.\] The two inequalities prove both equivalence and equality of norms. ◻

Proof. Let \(N_{12}\) be the number of \(p_2\)-clusters containing an infinite \(p_1\)-cluster. First, every automorphism-invariant event of the label field has probability zero or one. Indeed, approximate such an event \(E\), within error \(\varepsilon\) in probability, by a finite-coordinate event \(A\). Its coordinates lie in a finite ball centered at a vertex in an unbounded orbit; such an orbit exists because there are finitely many orbits and infinitely many vertices. An automorphism can move this ball disjointly from itself. The events \(A\) and its translate are then independent, whereas \(E\) agrees with its translate. It follows that \(\lvert\mathbb P(E)-\mathbb P(E)^2\rvert\le4\varepsilon\), and letting \(\varepsilon\downarrow0\) proves the assertion. In particular the invariant, measurable random variable \(N_{12}\in\{0,1,2,\ldots,\infty\}\) is constant almost surely. Since \(p_1>{p_c}\), an infinite \(p_1\)-cluster exists with positive probability, so this constant is positive.

It cannot equal one. Write \(\theta(x)=\mathbb P_{p_1}(|C_x|=\infty)\). Some \(\theta(x)\) is positive; positive association with a finite path makes \(\theta\) positive at every vertex. Invariance and the finite number of types give \(\theta_*:=\inf_x\theta(x)>0\). If \(N_{12}=1\), any two vertices in infinite \(p_1\)-clusters belong to the same \(p_2\)-cluster. Positive association therefore implies \[T_{p_2}(x,y)\ge\theta(x)\theta(y)\ge\theta_*^2 \qquad(x,y\in V).\] For finite sets \(F\) of arbitrarily large size this gives \(\langle\mathbf 1_F,T_{p_2}\mathbf 1_F\rangle/\lVert\mathbf 1_F\rVert_2^2 \ge\theta_*^2|F|\), contradicting boundedness.

Suppose instead that \(N_{12}=n\) almost surely for an integer \(n\ge2\). By countability there are fixed vertices \(x,y\) which, with positive probability, belong to infinite \(p_1\)-clusters lying in distinct \(p_2\)-clusters. Fix a finite path from \(x\) to \(y\). Replace its edge labels by independent uniform labels in \([0,p_1]\). The resulting law is absolutely continuous with respect to the original product law: only finitely many coordinates were resampled, each with a bounded density. At both parameters this modification only opens edges, and on the specified positive-probability event it merges two counted \(p_2\)-clusters.

Every infinite \(p_1\)-cluster after the modification contains an infinite \(p_1\)-cluster from before it. Indeed finitely many added edges can combine only finitely many old clusters into any new cluster; a finite union of finite clusters is finite. Thus each new counted \(p_2\)-cluster contains an old counted \(p_2\)-cluster, and old \(p_2\)-clusters only merge. On the event in question the new count is at most \(n-1\). This contradicts absolute continuity and the almost sure equality \(N_{12}=n\). Hence \(N_{12}=\infty\) almost surely.

On this single event, choose an infinite \(p_1\)-cluster in each of infinitely many distinct counted \(p_2\)-clusters. For every \(p\in[p_1,p_2]\), their containing \(p\)-clusters are infinite and remain distinct because their \(p_2\)-clusters are distinct. This proves the simultaneous assertion. ◻

Proof of 1. First suppose \(G\) is simple. If \(\mathop{\mathrm{Aut}}(G)\) is unimodular, 2 gives \(\lVert T_{{p_c}}\rVert<\infty\). Since \({p_c}<1\), sprinkling (13) at \(p={p_c}\) with any sufficiently small \(a>0\) gives a bounded connection kernel at \({p_c}+(1-{p_c})a>{p_c}\). Thus \({p_c}<{p_{2\to 2}}\).

If \(\mathop{\mathrm{Aut}}(G)\) is nonunimodular, 1 supplies both critical boundedness and the strict operator gap. In either case, 4 yields \({p_c}<{p_{2\to 2}}\le{p_u}\).

If loops or parallel edges are allowed, use the simple subdivision \(H\) from 1. Infinite-cluster counts in \(H\) at parameter \(t\) agree with those in \(G\) at parameter \(t^2\), so \({p_c}(H)^2={p_c}(G)\). Moreover, on the old vertex set, \[T^H_t\big|_{V(G)\times V(G)}=T^G_{t^2}.\] Compression of a bounded operator to this coordinate subspace is bounded. Applying the simple-graph result to \(H\) therefore gives boundedness of \(T^G_{{p_c}(G)}\) and of \(T^G_p\) at some \(p>{p_c}(G)\). The inequality \({p_{2\to 2}}(G)\le{p_u}(G)\) again follows from 4. This covers the full graph class in the statement.

Finally choose \({p_c}<p_1<p_2<1\) with \(T_{p_2}\) bounded, which the strict operator gap permits. 9 supplies the claimed simultaneous nonempty interval of infinitely many infinite clusters. The equality of the uniform subcritical and critical norms is 32, completing the proof. ◻

Proof of Corollary 2. For every graph in the corollary, 1 makes \(T_{{p_c}}\) a bounded operator on counting-measure \(\ell^2(V)\). Tonelli’s theorem identifies its nonnegative matrix product with the triangle sum. If \(\delta_v\) is the unit vector at \(v\), then \[\nabla_{{p_c}}(v) =\langle\delta_v,T_{{p_c}}^{3}\delta_v\rangle \le\lVert T_{{p_c}}^{3}\rVert_{2\to2} \le\lVert T_{{p_c}}\rVert_{2\to2}^{3}.\] This proves (1). The same implication is recorded in (Hutchcroft 2020d, sec. 1).