The two subexponential arguments

For a vertex \(x\), let \(B(x,r)\) be the graph-distance ball of integer radius \(r\). Put \(V_-(r)=\min_x|B(x,r)|\) and \(V_+(r)=\max_x|B(x,r)|\). Quasi-transitivity makes these extrema finite and compares them up to a bounded shift of \(r\). It also implies that \(V_+(r)^{1/r}\) has a limit. A limit greater than one is the exponential case covered by Hutchcroft. In the other case, the proof divides according to the following exhaustive alternatives:

1. \(\log V_-(r)/\log r\to\infty\);

2. \(V_-(r)\le r^d\) along an unbounded sequence of radii, for some fixed finite \(d\).

This division requires no classification of intermediate growth.

In case (i), suppose that an infinite cluster occurs at \(p_c\). At a slightly smaller parameter, large finite clusters still occur with probability bounded away from zero. The volume growth then places two distinct large clusters within a distance that is small on the logarithmic scale of their sizes. Repeatedly bisecting a path between them produces a single adjacent pair. Each bisection adds at most one cluster and one possible connecting edge. We choose these edges using the decrease in a search’s conditional success probability, and control the total cost by relative entropy. Section 3 establishes the search and entropy estimates. Sections 4 and 5 show that the resulting adjacent pair contradicts the bound in Proposition 4: across a fixed edge, distinct clusters each of size at least \(s\) occur with probability at most \(Cs^{-1/2}\).

In case (ii), we apply the finitary structure theorem of Tessera and Tointon (Tessera and Tointon 2021, Corollary 2.4) to an auxiliary vertex-transitive graph. Lifting its quotient action to \(G\) and passing to a finite-index subgroup gives a finitely generated nilpotent quotient and a kernel with finite vertex orbits. Section 7 uses an induction on the number of infinite cyclic factors in a chosen series of that group. The one-dimensional case has \(p_c=1\). Otherwise there are two integer coordinates on vertices that change by a fixed translation under each group element. The induction excludes an infinite open ray whose two coordinates both remain bounded.

Under the assumed critical percolation, uniqueness and this exclusion produce two high-probability approximate moves in linearly independent coordinate directions in Section 8. After rescaling, the moves advance by \((1,1)\) and \((1,-1)\), with arbitrarily small edge steps. They persist at a parameter below \(p_c\). In Section 9, we place corridors in these coordinates, with intersections confined to their common end regions. The joint gluing inequality (Theorem 10) transfers connections through each fresh corridor: reaching a relay set whose vertices have uniformly high target probabilities gives a high conditional chance of continuing to the target. Finally an adaptive exploration keeps the conditional chance of failure small at every tested site. A planar boundary count gives positive probability of infinitely many successful sites, hence an infinite open cluster below \(p_c\). All paths and all revealed edges belong to \(G\); the quotient supplies coordinates for this construction.

History and related methods

Percolation was introduced by Broadbent and Hammersley (Broadbent and Hammersley 1957). For nearest-neighbor bond percolation on \(\mathbb Z^d\), planar methods established critical nonpercolation in dimension two (Harris 1960; Kesten 1980). The lace expansion gave the result in sufficiently high dimensions (Hara and Slade 1990), and Fitzner and van der Hofstad obtained it for \(d\ge11\) (Fitzner and Hofstad 2017). The sharpness theorems of Aizenman–Barsky and Menshikov describe the subcritical regime (Aizenman and Barsky 1987; Men’shikov 1986); they do not by themselves settle the behavior at the threshold.

Beyond lattices, Benjamini, Lyons, Peres, and Schramm proved critical nonpercolation for nonamenable graphs admitting a unimodular quasi-transitive action (Benjamini et al. 1999). Timár excluded infinitely many infinite critical clusters on transitive nonunimodular graphs (Timár 2006), leaving the possibility of a unique infinite cluster. Hutchcroft excluded all infinite critical clusters on every quasi-transitive graph of exponential growth (Hutchcroft 2016, Theorem 1.2); we use that entire theorem. Hermon and Hutchcroft treated unimodular quasi-transitive graphs satisfying a uniform stretched-exponential return-probability bound with exponent greater than \(1/2\) (Hermon and Hutchcroft 2021, Theorem 1.2). The two arguments above cover the remaining subexponential case.

Our joint gluing theorem also has a finite-graph and lattice history that is distinct from the general quasi-transitive conclusion. Kozma and Nitzan conjectured the multiplicative connection inequality (Kozma and Nitzan 2024, Conjecture 1) and showed that their weaker uniform near-one gluing Conjecture 3 implies critical nonpercolation on \(\mathbb Z^d\) for every \(d\ge2\) (Kozma and Nitzan 2024, Theorem 6). Leder’s public account reports a Lean formalization of that lattice conclusion and displays the first-contact comparison (GEN) (Leder 2026, sec. 3.4). Although the account does not claim Conjecture 1, (GEN) implies the finite joint inequality: take its increasing cluster functional to be the indicator of meeting the target and bound the resulting weighted mean by its minimum. The subsequent public manuscript (ChatGPT 5.6 Sol and Claude Fable 5.1 2026) explicitly records this consequence and stronger finite connection comparisons. Its public site-percolation extension (ChatGPT 5.6 2026) treats independent labelled hyperedges and critical site percolation on \(\mathbb Z^d\) for \(d\ge3\). We give a direct joint-gluing proof with this antecedent credit. These finite and lattice results do not assert Theorem 1 for general quasi-transitive graphs.

The probabilistic tools are developed with their local predecessors at their points of use. The adjacent-cluster bound adapts the boundary-discrepancy method of Aizenman, Kesten, and Newman (Aizenman et al. 1987, sec. 3) and Hutchcroft’s ghost-field formulation (Hutchcroft 2020, sec. 3). Conditional association was developed by van den Berg and Kahn (Berg and Kahn 2001); alternating conditional-cluster resampling is the method of van den Berg, Häggström, and Kahn (Berg et al. 2006). Decision-tree inequalities (O’Donnell et al. 2005; Duminil-Copin et al. 2019) provide related links between exploration and pivotality; here the local estimate uses the martingale of conditional connection probabilities. The entropy bound accommodates adaptive read sets whose overlap is controlled only under a comparison law, beyond the fixed-set setting of Shearer’s inequality and its relative-entropy forms (Chung et al. 1986; Madiman and Tetali 2010).

The geometric construction adapts the seed and adaptive-block strategy of Grimmett and Marstrand (Grimmett and Marstrand 1990) and the fresh-collar and stored-prediction arguments of Kozma and Nitzan (Kozma and Nitzan 2024, Lemma 10 and proof of Theorem 6). Critical extinction in each fixed slab (Duminil-Copin et al. 2016) does not alone give the full-lattice conclusion. Choosing useful directions by exit probabilities is related to Martineau and Tassion’s parallelogram construction (Martineau and Tassion 2017). Here the two-direction argument and the conditional-failure boundary count are carried out on the original graph, including when the reveal domains are countable.

The structural reduction runs from Gromov’s polynomial-growth theorem (Gromov 1981) through Trofimov’s graph and near-polynomial-growth theorems (Trofimov 1985, 2003) to Tessera and Tointon’s one-scale theorem (Tessera and Tointon 2021, Corollary 2.4). The latter develops the approximate-group method of Breuillard, Green, and Tao (Breuillard et al. 2012) and supplies the quotient action and stabilizer information used here. Related nilpotent quotients occur in percolation locality on polynomial-growth graphs (Contreras et al. 2023). In this paper the quotient supplies coordinates; every connection and exposure uses original edges.

Graph conventions and the threshold

It suffices to prove Theorem 1 for simple graphs. Indeed, loops have no effect on clusters. If parallel edges occur, subdivide each edge once, using a distinct new vertex for each edge. The resulting graph is simple and remains connected and locally finite. There are finitely many edge orbits, since there are finitely many vertex orbits and each representative has finite degree. Lifting the original automorphisms therefore proves quasi-transitivity of the subdivided graph. At parameter \(t\), each original edge is fully traversable with probability \(t^2\), independently of the others. An infinite cluster in the subdivided graph contains infinitely many original vertices, by local finiteness. Thus percolation on it occurs exactly when the effective configuration on the original graph percolates, and its critical probability is \(\sqrt{p_c(G)}\). The simple-graph conclusion at this parameter gives the desired conclusion on \(G\).

Henceforth the infinite graph is simple. Finite auxiliary graphs may have parallel edges after contractions. Local finiteness and connectedness imply countability. Finitely many vertex orbits imply \(D<\infty\). For \(p<1/D\), the probability of an open simple path of length \(n\) starting at \(x\) is at most \(D^np^n\), which tends to zero. An infinite cluster would contain such a path for every \(n\), so \[ p_c\ge 1/D>0. \tag{1}\]

The event that some infinite cluster exists is unchanged by altering finitely many edge states: adding finitely many edges joins only finitely many clusters, and deleting finitely many edges splits an infinite cluster into finitely many pieces. It is therefore a tail event and has probability zero or one. Monotonicity in \(p\) shows that our definition of \(p_c\) agrees with the supremum of the parameters at which almost surely every cluster is finite. In particular all clusters are finite almost surely for \(p<p_c\).

We use Harris’s inequality: increasing functions of independent edge states have nonnegative covariance (Harris 1960). For finitely many bits, this follows by induction from the covariance decomposition after conditioning on one bit; the two conditional means are themselves increasing. For bounded functions of countably many bits, conditional expectations on finite sets of coordinates and martingale convergence give the same conclusion. Decreasing functions are likewise positively correlated; an increasing and a decreasing function are negatively correlated.

If critical percolation occurs with positive probability, countability gives a vertex \(x\) with \(\mathbb P_{p_c}(|C_x|=\infty)>0\). For a fixed path from \(y\) to \(x\), Harris’s inequality bounds \(\mathbb P_{p_c}(|C_y|=\infty)\) below by this positive probability times \(p_c\) to the path length. Finitely many vertex orbits then give \[ \theta:=\min_{x\in V}\mathbb P_{p_c}(|C_x|=\infty)>0. \tag{2}\] This uniform lower bound is the contradiction hypothesis in both subexponential arguments.

Ball comparisons

We now invoke precisely the following external result.

It remains to work under \(\log V_+(r)=o(r)\). Either \[ \frac{\log V_-(r)}{\log r}\longrightarrow\infty, \tag{4}\] or this quotient is bounded along an unbounded sequence of integer radii. In the latter case there is a finite \(d\) such that \[ V_-(r)\le r^d \quad\text{along that sequence.} \tag{5}\] These alternatives are exhaustive, including when the quotient oscillates.

Exact clusters and finite witnesses

Conditioning on the exact vertex set of a cluster fixes its boundary edges closed and imposes only an internal connectivity condition. Consequently the edges induced outside that set keep their product law. Revealing every incident edge, including internal edges, has the same property. We will specify which of these two exposures is being used.

The event \(|C_x|\ge s\), for a fixed positive integer \(s\), can be witnessed inside \(B(x,s-1)\): a connected cluster of at least \(s\) vertices contains a connected set of \(s\) vertices containing \(x\). Its probability is therefore continuous in \(p\). A connection between fixed finite sets is the increasing union of connections in finite induced subgraphs. Its probability can be approximated from below by such finite events, at the same parameter \(p\). These observations also apply uniformly to finitely many vertex or pair orbits.

Adjacent large distinct clusters

For an edge \(e=xy\) and an integer \(s\ge1\), write \[H_s(e)=\{C_x\ne C_y,\ |C_x|\ge s,\ |C_y|\ge s\}.\] Cluster size means the number of vertices. The next estimate is uniform over the edges, including those in different automorphism orbits. Its proof adapts the cluster-boundary discrepancy argument of Aizenman, Kesten, and Newman (Aizenman et al. 1987, sec. 3) and Hutchcroft’s ghost-field and martingale formulation (Hutchcroft 2020, sec. 3). We give the finite-graph averaging argument and its quasi-transitive passage explicitly.

Clusters conditioned to avoid one another

Conditional association of connection events was established by van den Berg and Kahn (Berg and Kahn 2001). The pivotal estimate uses the cluster-function correlation inequality of van den Berg, Häggström, and Kahn (Berg et al. 2006, Theorems 1.3 and 1.4). We include the product-measure proof by alternating cluster updates, the method of (Berg et al. 2006, sec. 2.1). Here a cluster object records its vertices and all its open edges. For a root set, take the union of the cluster objects of its roots. Objects are ordered by inclusion of both vertices and edges. A random object is associated if any two increasing real-valued functions of it have nonnegative covariance.

A depth-first pivotal second moment

Fix a finite undirected graph, a vertex \(w\), and a deterministic target set \(B\). Write \(E=\{w\leftrightarrow B\}\) and let \(\mu\) be homogeneous Bernoulli percolation with parameter \(0<p<1\), with \(q=1-p\). Parallel edges are allowed. We use the following deterministic lazy depth-first search. On reaching a vertex, push its unqueried edges to unreached vertices onto a stack in a fixed order. Pop the last entry; if its head has meanwhile been reached, skip it without reading its bit. Otherwise query the bit, reaching its head when it is open. Stop on reaching \(B\) or exhausting the stack.

A query prefix records the previously queried edges and their states. At a query of \(e\), put \[h=\mu(E\mid\text{prefix}),\qquad h_b=\mu(E\mid\text{prefix},X_e=b),\qquad d_e=h_1-h_0.\] These probabilities condition only on that search’s own query prefix. They are not conditioned on any additional cluster information that another construction may possess. Since \(E\) is increasing, \(0\le d_e\le1\). A queried edge is an actual open pivot if it is open and closing it destroys \(E\) in the full configuration, including unqueried edges.

Entropy of overlapping transcripts

For fixed coordinate subsets, related bounds are given by Shearer’s inequality (Chung et al. 1986) and its relative-entropy forms (Madiman and Tetali 2010, sec. IX). The read sets below are adaptive, and their overlap is bounded only under a possibly dependent comparison law. We prove the required estimate by ordering the algorithms randomly and charging each freshly queried bit to a common transcript.

A transcript records an algorithm’s ordered queries and answers, including its stopping decision. The next result charges several such transcripts to the information in one common configuration. Its overlap hypothesis is required only under the law being estimated.

For probability laws \(Q,R\) on a finite set, let \[D(Q\Vert R)=\sum_x Q(x)\log\frac{Q(x)}{R(x)}\] be their relative entropy, with zero summands omitted.

Contact weights and normalized exposures

We first record the normalization that makes the construction possible. Suppose a union \(O\) of exact clusters has been exposed, a deterministic target set \(\mathcal A\) is contained in \(O\), and \(w\notin O\). Run the deterministic lazy depth-first search of Lemma 6 from \(w\) towards \(\mathcal A\). For every query \(f\), let \(h\) be the conditional probability of \(w\leftrightarrow\mathcal A\) under the original product law \(\mu\), conditioned only on this search’s own query prefix. Write \(h_0,h_1\) for the two continuations and \(d_f=h_1-h_0\ge0\). Thus \(h=ph_1+qh_0\).

A contact is a queried edge from the newly explored cluster into \(O\). Every contact is closed in the exposed configuration. Give it weight \[a_f=pd_f, \qquad A=\sum_{\text{queried contacts }f}a_f, \qquad h_* =\mu(w\leftrightarrow\mathcal A).\] The numbers \(d_f\) are not recomputed under the law conditioned on \(O\). This distinction is essential: under that conditioned law a contact is forced closed, and the ordinary conditional probability drops by \(h-h_0=pd_f\). Every noncontact query is a fresh Bernoulli bit, so its conditional expected change in \(h\) is zero. The search fails and its final prefix certifies failure even under \(\mu\), since it has queried all edges leaving the cluster of \(w\). Telescoping therefore gives \[ \mathbb E_{\mathrm{ext}}A=h_*. \tag{20}\] Here the expectation samples only the product exterior of the fixed incidence exposure.

Conditional on the fixed incidence exposure and any query prefix, the same telescoping gives expected remaining contact weight at most one. The potential \(h\) still uses only the search’s own prefix. Since each contact weight is at most one, expansion of the square yields \[ \mathbb E_{\mathrm{ext}}A^2 \le \mathbb E_{\mathrm{ext}}\sum_f a_f +2\mathbb E_{\mathrm{ext}}\sum_f a_f \le3. \tag{21}\] In the middle expression the second sum bounds each contact weight multiplied by the conditional expected sum of subsequent weights. Consequently, when \(h_*>0\), the rule \[ \frac{a_f}{h_*}\,\mu_{\mathrm{ext}}(d\omega) \quad\text{on exterior configurations and a chosen contact }f \tag{22}\] defines a probability measure, with counting measure on the contact coordinate.

After this choice, expose the exact incidence configuration of the new cluster \(K=C_w\). The chosen contact, its query prefix, and its weight are determined by the old exposure and the incidence states of \(K\). Thus the factor in (22) imposes no condition on edges wholly outside \(O\cup K\). Conditional on the new exposure and chosen contact, those edges still have product law. This proves both normalization and preservation of the exterior law at each successive exposure.

Binary interpolation and its description

Let \(J=\lceil\log_2d\rceil\). Start with the product law conditioned on the endpoint event in Proposition 8. Expose the two endpoint clusters, calling them the left and right bases, and assign every vertex in each base its respective side. Maintain an interval of the path whose endpoints have opposite sides. At a step \(i\), take its midpoint \(w_i\), using the lower index in a tie, and set \[\mathcal A_i=\{x_0,x_d,w_1,\ldots,w_{i-1}\}.\] Every target anchor belongs to a previously exposed cluster. If \(w_i\) already lies in their union \(O\), make no new exposure and give it its cluster’s side; call this a hold. Otherwise apply (22) with \(w=w_i\) and \(\mathcal A=\mathcal A_i\), expose \(K_i=C_{w_i}\), and record its chosen contact \(f_i\). The cluster containing the head of \(f_i\) is its parent, and \(K_i\) inherits that parent’s side. Leave \(f_i\) closed. Retain the subinterval joining \(w_i\) to the endpoint of opposite side.

The new interval has length at most the ceiling of half the old length, so after \(\ell\le J\) steps its endpoints are adjacent. At each nonhold step, one current endpoint is a target anchor at path-index distance strictly less than \(d\) from \(w_i\). Therefore \[ h_i:=\mu(w_i\leftrightarrow\mathcal A_i)\ge c. \tag{23}\] The preceding normalization constructs a joint law of a common final configuration \(\Omega\) and its chosen contacts. Equivalently, one successively multiplies the conditional exterior law by the normalized factors (22); each factor is measurable in the new frozen incidence data, so later exterior sampling preserves all earlier factors.

The parent relations form a forest rooted at the two bases. Call a nonbase cluster active if it lies on the ancestry path from the owner of one terminal endpoint to its base. The active clusters on each side form one chain. Record a descriptor consisting of the number of steps, the side and hold bit at each step, and the active indices. There are at most \(\sum_{j=0}^J8^j\le8^{J+1}\) descriptors. Condition on one of maximum probability and denote the resulting joint law by \(Q\). All midpoints, target sets, nonhold indices \(I\), active indices \(I_*\), and the terminal edge are now deterministic. Moreover, the parent of an active index is the preceding active index on its side, or its base if there is none. Indeed, the active indices on that side are exactly one ancestry chain, ordered by creation time. Thus every active parent has a deterministic anchor.

Relative to \(\mu\) on configurations and counting measure on the contacts indexed by \(I\), the joint density satisfies \[ \frac{dQ}{d(\mu\otimes\mathrm{count})} \le e^L\prod_{i\in I}a_i, \qquad a_i=pd_{f_i}, \qquad L=\log(1/\delta)+(J+1)\log8+J\log(1/c). \tag{24}\] The three costs are the endpoint conditioning, descriptor conditioning, and the normalizers (23).

The two chains and their inverse

Open only the active contacts in \(\Omega\), and call the resulting configuration \(\eta\). Each terminal endpoint is now joined along its active ancestry chain to a base cluster of size at least \(s\). The two resulting clusters remain distinct: all original cluster boundaries were closed, and the only newly opened edges join clusters within one of the two same-side chains. Thus \(Q_\eta\) is supported on \(H_s(e)\) for the terminal edge fixed by the descriptor.

We next give a reverse procedure and compute its probability of recovering the chosen contacts. This computation will identify the quantities that the remaining entropy estimates must control.

For the fixed schedule, expose all anchor clusters in order on an ordinary product configuration. Let \(A_i\) be the sum of the contact weights at step \(i\), with \(A_i=1\) when \(w_i\) is already exposed. These quantities depend only on the configuration and schedule, not on earlier contact choices: the exposed union is exactly the union of the clusters of the preceding anchors. Write \[X=\sum_{i\in I}\log^+A_i, \qquad \log^+t=\log\max\{1,t\}.\]

For each active \(i\), let \(t_i\) be a standalone deterministic transcript that exposes the two exact clusters at \(w_i\) and its parent anchor, querying all their incidence edges and caching duplicate queries. For an ordinary pair transcript \(t\), call a contact \(f\) admissible if the cluster at \(w_i\) avoids \(\mathcal A_i\) and its depth-first search queries \(f\) closed into the distinct parent cluster. Every pair selected under \(Q\) is admissible.

Open just \(f\) in an admissible configuration. It becomes the sole open exit from the original cluster of \(w_i\), and the parent contains a target anchor. Hence \(f\) is a queried actual open pivot. Let \(S_i(t,f)\) be the sum of the weights \(qd\) of queried actual open pivots through \(f\) in this modified search, including \(f\) itself. This number is determined by \((t,f)\). The prefix through \(f\) uses only incidences of the original cluster at \(w_i\); an earlier queried open edge is an actual pivot exactly when deleting it separates \(w_i\) from the tail of \(f\) within that cluster. The exact incidence transcript records all the internal edges needed for this test.

Construct a reference probability \(R\) on output configurations and marks as follows. First sample \(\eta\) under \(\mu\). On a working copy, process the active indices in descending order. At index \(i\), run the \(\mathcal A_i\) depth-first search from \(w_i\). For each queried actual open pivot of weight \(b_f=qd_f\), let \(S_f\) be the cumulative weight through it, and give that pivot guess probability \[ \pi_i(f)=\frac{b_f}{(1+S_f)^2}. \tag{25}\] The sum of these probabilities is at most \(\int_0^\infty(1+t)^{-2}\,dt=1\), by comparison with the integral on each cumulative-weight interval. Assign the remaining probability to a dummy guess. Close the guessed pivot and continue, using arbitrary dummy defaults after an invalid history. After all active guesses, on the recovered working copy guess each inactive contact independently with probabilities \(a_f/A_i\), again using a dummy default if necessary. The configuration marginal of \(R\) is \(\mu\).

We verify that the true inverse guesses have precisely the weights just defined. At active step \(i\), all later active contacts have already been closed. Every earlier active contact has both endpoints in original clusters created before \(K_i\), since a contact joins its new cluster to an older parent. Thus none of those earlier contacts touches \(K_i\). Its only open boundary edge in the current working copy is \(f_i\), and its parent still contains the deterministic target anchor. The search prefix through \(f_i\), its original-product conditional differences \(d_f\), and the criterion for earlier actual pivots are therefore unchanged from the configuration with only \(f_i\) opened. In particular, the guess \(f_i\) is available with cumulative weight exactly \(S_i\). Holds add target anchors but no new cluster or switched edge, so they do not affect this argument.

The active contacts are distinct: different nonhold indices create distinct original clusters, and each contact joins its new cluster to an older one. Consequently the map \((\Omega,(f_i))\mapsto(\eta,(f_i))\) is injective. Along the true inverse history, (24) bounds the ratio of the joint output density to \(R\) by \[e^L\prod_{i\in I_*} \frac{(q/p)a_i}{qd_{f_i}/(1+S_i)^2} \prod_{i\in I\setminus I_*}A_i =e^L\prod_{i\in I_*}(1+S_i)^2 \prod_{i\in I\setminus I_*}A_i.\] Here \(q/p\) is the original-to-opened Bernoulli likelihood ratio for each active contact. It cancels because the forward weight is \(a_i=pd_{f_i}\) and the inverse guess has numerator \(qd_{f_i}\). Taking logarithms and discarding the marks by data processing gives \[ D(Q_\eta\Vert\mu) \le L+2\mathbb E_Q\sum_{i\in I_*}\log(1+S_i)+\mathbb E_Q X. \tag{26}\] It remains to bound the two expectations on the right by a constant times \(L+J\). We do this under the closed-contact law \(Q\), before opening any edge.

Entropy of the sampled configuration

We use the same formula for \(D(P\Vert R)\) when \(R\) is a nonnegative finite measure rather than a probability; in that case it need not be nonnegative. For \(P\ll R\), Jensen’s inequality gives \[ \mathbb E_P g\le D(P\Vert R)+\log\sum R e^g. \tag{27}\] Write \(Q_\Omega\) for the configuration marginal of \(Q\) and put \(D_\Omega=D(Q_\Omega\Vert\mu)\).

By (21), \[\mathbb E_\mu[\max(1,A_i)^2\mid\text{previous exact exposures}]\le4.\] Each \(A_i\) is known after the next exact exposure. Successive conditional expectation therefore gives \(\mathbb E_\mu e^{2X}\le4^J\). Summing the density bound (24) over the possible contact choices, which only enlarges the sum if the descriptor restrictions are dropped, gives \[D_\Omega\le L+\mathbb E_Q X.\] Applying (27) to \(2X\) gives \(2\mathbb E_Q X\le D_\Omega+J\log4\). Hence \[ D_\Omega\le2L+J\log4, \qquad \mathbb E_Q X\le L+J\log4. \tag{28}\]

Entropy of the inverse guesses

We now estimate the sum involving \(S_i\). The pair transcripts \(t_i\) contain just the cluster data needed to determine each marked weight. On the support of \(Q\), every cluster occurs in at most two such pairs, and an edge is incident to at most two clusters. Each bit is therefore read by at most four pair algorithms. Lemma 7 gives \[ \sum_{i\in I_*}D(Q_{t_i}\Vert\mu_{t_i})\le C_pD_\Omega, \tag{29}\] with \(C_p\) uniform when \(p,q\) are bounded below. The multiplicity condition is needed only on the support of \(Q\).

For each active \(i\), define a finite measure on marked pair transcripts by \[\nu_i(t,f)=\mu_{t_i}(t)\,pd_f \quad\text{for admissible }(t,f),\] and zero otherwise. This reference measure combines the original pair transcript law with the contact weight used in the forward sampling. It need not have mass one. Its usefulness is the following bound, which converts the cost of an inverse guess to the pivotal second moment.

The map that opens \(f\) and records \(f\) is injective on full marked configurations. The original-to-modified product probability ratio is \(q/p\), so multiplying the original weight \(pd_f\) gives the modified weight \(qd_f\). In any modified configuration, the admissible inverse marks are a subset of its queried actual open pivots. Their cumulative weights are at most the total \(W\) from Lemma 6. Therefore \[ \sum_{t,f}\nu_i(t,f)(1+S_i(t,f)) \le\mathbb E_\mu[W(1+W)]\le4. \tag{30}\]

We next control the entropy of the chosen marks, not just that of their pair transcripts. Write \(\mathcal H(\cdot\mid\cdot)\) for conditional Shannon entropy. Taking logarithms in (24) and averaging yields \[ D_\Omega+\sum_{i\in I}\mathbb E_Q\log(1/a_i) -\mathcal H((f_i)_{i\in I}\mid\Omega)\le L. \tag{31}\] Since \(t_i\) is determined by \(\Omega\), \[\mathcal H((f_i)_{i\in I}\mid\Omega) \le\sum_{i\in I_*}\mathcal H(f_i\mid t_i) +\sum_{i\in I\setminus I_*}\mathcal H(f_i\mid\Omega).\] For an inactive index, nonnegativity of conditional relative entropy to the probabilities \(a_f/A_i\) gives \[\mathbb E_Q\log(1/a_i)-\mathcal H(f_i\mid\Omega) \ge-\mathbb E_Q\log A_i.\] Combining these inequalities with (28) and (29), we obtain \[\begin{align*} \sum_{i\in I_*}D(Q_{t_i,f_i}\Vert\nu_i) &=\sum_{i\in I_*}\left[ D(Q_{t_i}\Vert\mu_{t_i})+ \mathbb E_Q\log(1/a_i)-\mathcal H(f_i\mid t_i)\right]\\ &\le C_pD_\Omega+L+\mathbb E_Q X \le C'_p(L+J). \end{align*}\] All chosen weights are positive, and all spaces are finite, so these expressions are finite on the support of \(Q\). Apply (27) with \(g=\log(1+S_i)\) and use (30). This proves \[ \mathbb E_Q\sum_{i\in I_*}\log(1+S_i)\le C''_p(L+J). \tag{32}\]

Substituting (28) and (32) into (26) yields \[ D(Q_\eta\Vert\mu)\le C'''_p(L+J). \tag{33}\] We already proved that \(Q_\eta\) is supported on \(H_s(e)\). If \(u=\mu(H_s(e))\), conditioning \(\mu\) on this event gives \(D(Q_\eta\Vert\mu)\ge\log(1/u)\). Together with the definition of \(L\), the last display proves Proposition 8.

Conditional cluster columns

Throughout the finite-graph proof assume \(p_e<1\) for every edge; zero probabilities are permitted. If \(C(S)=w\) has positive probability, its occurrence is determined by edges having an endpoint in \(w\): all vertices of \(w\) must connect to \(S\) within \(w\), and all edges from \(w\) to its complement must be closed. Thus edges induced outside \(w\) retain their independent original laws. The same observation applies inside a graph obtained by vertex deletion. For \(S=\varnothing\), its cluster is empty.

Choose distinct distinguished vertices \(x_1,\ldots,x_N\). They need not exhaust the vertices of \(G\); all other vertices and their edges remain in the graph. For \(1\le k\le N\), put \[S_k=\{x_1,\ldots,x_{k-1}\},\qquad E_k=\{C(x_k)\cap S_k=\varnothing\},\qquad d_k=\mathbb P(E_k)>0.\] The positivity follows by closing every edge incident to \(x_k\). Let \(\mu_k\) be the law of \(C(x_k)\) conditional on \(E_k\), and set \[\mathcal D_k=\{U\subseteq V(G):x_k\in U,\ U\cap S_k=\varnothing\}.\] For \(F:\mathcal D_k\to[0,\infty)\), define vectors in \(\mathbb R^N\) by \[v_k^F(i)=\mathbb E_{\mu_k} [\mathbf1_{\{x_i\in C(x_k)\}}F(C(x_k))], \qquad h_k=v_k^1.\] The matrix \(H=(h_1\ \cdots\ h_N)\) is unit lower triangular.

The vector \(v_k^F-(\mathbb E_{\mu_k}F)h_k\) records the conditional covariances of \(F(C(x_k))\) with the distinguished-vertex indicators. For increasing \(F\), we will show that this vector is a nonnegative combination of later columns. The following deletion estimate supplies the nonnegative terms in that combination.

The deletion estimate and column positivity

Nonnegative relay weights

Apply Lemma 12 to \[F(U)=\mathbf1_{\{U\cap\{x_{k+1},\ldots,x_N\}\ne\varnothing\}}.\] Its mean \(m_k\) under \(\mu_k\) is less than one, since \(x_k\) can be isolated. The centered vector is \((1-m_k)(h_k-e_k)\), where \(e_k\) is the \(k\)th coordinate vector. Hence there are numbers \(\Gamma_{lk}\ge0\), \(l>k\), such that \[h_k=e_k+\sum_{l>k}\Gamma_{lk}h_l.\] Let \(\Gamma\) be zero on and above the diagonal. In matrix form, \(H=I+H\Gamma\), and therefore \[ H^{-1}=I-\Gamma. \tag{44}\] Moreover, partitioning the cluster of \(x_i\) by its first distinguished vertex gives \[ Hd=\mathbf1,\qquad (I-\Gamma)\mathbf1=d, \qquad d=(d_1,\ldots,d_N)^{\mathsf T}. \tag{45}\]

The theorem and corollary may be applied after fixing any collection of edge states, provided the remaining edges still have product law. Relay and target sets must then be fixed under that conditioning. In particular, a later adaptive choice of contacts is not an additional conditioning to be inserted into their connection probabilities.

We apply this inequality in the corridor construction of Section 9. The remaining sections develop the geometric argument for graphs satisfying the polynomial bound along an unbounded sequence of radii, beginning with the quotient action that will supply the corridor coordinates.

From polynomial scales to the quotient action

The structural ancestry is Gromov’s group theorem (Gromov 1981), Trofimov’s graph and near-polynomial growth theorems (Trofimov 1985, 2003), and Tessera and Tointon’s one-scale form (Tessera and Tointon 2021). The proof below uses the latter with its action and stabilizer conclusions.

Cyclic series and the decreasing parameter

A cyclic series is a finite subnormal series \[N=N_0\triangleright N_1\triangleright\cdots\triangleright N_k=\{1\}\] whose factors \(N_i/N_{i+1}\) are cyclic, possibly finite or trivial. We use the number of its infinite factors. The induction applies to all data equipped with a chosen series of at most a given count; independence of that count from the choice of series is not needed.

The group facts below are classical; see (Robinson 1996, Theorems 5.2.5 and 5.2.18). We give the arguments to make the decrease of the induction parameter explicit.

The induction and equivariant coordinates

Fix action data \((G,\Lambda,\varrho,N)\) as in Theorem 14, together with a cyclic series having \(h\) infinite factors. We prove the theorem simultaneously for all such data by induction on \(h\). A series with no infinite factor makes \(N\) finite. Each \(\Lambda\)-orbit is a finite union of \(\ker\varrho\)-orbits and is therefore finite. There are finitely many \(\Lambda\)-orbits, contrary to infinitude of \(G\). This is the base case. For the induction step, assume the theorem for every such action whose quotient has a cyclic series with fewer than \(h\) infinite factors. Suppose, towards a contradiction, that \(p_c(G)<1\) and critical percolation has an infinite cluster with positive probability.

We first construct the coordinates and dispose of the one-dimensional case. Suppose \(\phi:N\to\mathbb Z^j\) is an epimorphism, where \(j=1\) or \(2\). Choose representatives \(v_1,\ldots,v_k\) for the \(\Lambda\)-orbits and put \[ \pi(gv_i)=\phi\varrho(g),\qquad g\in\Lambda. \tag{46}\] This is well defined: if \(gv_i=g'v_i\), then \((g')^{-1}g\) stabilizes \(v_i\), its \(\varrho\)-image belongs to a finite group, and \(\phi\) annihilates that image because \(\mathbb Z^j\) is torsion free. Moreover, \[ \pi(gx)=\phi\varrho(g)+\pi(x). \tag{47}\] The map is onto, even on each vertex orbit. There are finitely many oriented edge orbits, so equivariance gives a finite bound \[ D_\pi:=\sup_{x\sim y}|\pi(x)-\pi(y)|<\infty. \tag{48}\] The norm is Euclidean when \(j=2\).

Suppose that \(\phi:N\to\mathbb Z\) has finite kernel. Then \(L_0=\ker(\phi\varrho)\) has finite orbits: its image under \(\varrho\) is finite, and its kernel has finite orbits. Thus \[\pi^{-1}(0)=\bigcup_{i=1}^k L_0v_i\] is finite. An element with \(\phi\varrho\)-value \(t\) bijects this level onto \(\pi^{-1}(t)\), so all integer levels have the same finite size.

Choose an integer \(J\ge\max(1,D_\pi)\). Edges crossing the cut from \(\{\pi\le t\}\) to \(\{\pi>t\}\) have endpoints in a fixed number of levels around \(t\). Their number is uniformly bounded, since degree and level sizes are bounded. The crossing-edge sets for levels \(t\in(J+1)\mathbb Z\) are disjoint. For every \(p<1\), their all-closed events are independent with probabilities bounded below by a positive constant. Almost surely infinitely many such cuts are closed in both directions. Every open cluster is then confined between two closed cuts and lies in finitely many finite levels. This proves \[ p_c(G)=1 \quad\text{whenever $N$ has an epimorphism to $\mathbb Z$ with finite kernel.} \tag{49}\]

Consequently, the hypotheses of Theorem 14 with \(p_c(G)<1\) force the abelianization of \(N\) to have rank at least two. It therefore admits an epimorphism onto \(\mathbb Z^2\).

Excluding rays with bounded projection

We now choose an epimorphism \(\phi:N\to\mathbb Z^2\), as justified above, and let \(\pi\) be the associated coordinate map. The smaller-count induction applies to a component over a bounded box because its preserving subgroup maps into \(\ker\phi\), an infinite-index subgroup of \(N\). The next lemma checks all the required action properties.

Thus the induction step has onto equivariant coordinates in \(\mathbb Z^2\), bounded coordinate jumps, and no infinite open ray trapped over a bounded set. The following two sections construct an infinite cluster at a parameter below \(p_c(G)\) from the assumed critical jump. That contradiction completes the induction and proves Theorem 14. Proposition 15 then applies it to the remaining growth case.

The deterministic layout

The constant \(C\) is the path-displacement bound from Proposition 18. Let \(B\) be the initial vertical tolerance for an entrance, and let \(H\) be the tube half-width. Fix \[C=100,\qquad \mu=\tfrac14,\qquad B=2C+10,\qquad H=3B.\] An endpoint rectangle of time half-width \(J\) will contain every overlap of incident tubes. Choose that width first, and then the corridor length, by taking integers \(J,L\) with \[J>10(H/\mu+C),\qquad L>10(J+B),\qquad T=L+\tfrac12.\] For a site \(v=(a,b)\in\mathbb N_0^2\), let \(c_v=((a+b)T,(a-b)\mu T)\). Its children are \((a+1,b)\) and \((a,b+1)\); their displacement from \(v\) is \((T,\sigma\mu T)\) with \(\sigma=1\) and \(-1\), respectively. All regions below mean vertex preimages under the coordinates \((t,z)\), with coordinates taken relative to the indicated center. Define \[Q_v=\{|t-c_v^{(1)}|\le J,\ |z-c_v^{(2)}|\le\mu J+H\}.\] For a parent–child pair \(v,x\) of sign \(\sigma\), let \(D_{vx}\) be the induced subgraph on the union of \(Q_v,Q_x\) and the tube \[0\le t-c_v^{(1)}\le T,\qquad |z-c_v^{(2)}-\sigma\mu(t-c_v^{(1)})|\le H.\] Write \(\mathcal E_{vx}\) for its edge set, and \(\mathcal Q_v\) for the edge set induced by \(Q_v\).

Relay regions and the order of scales

Inside a corridor we will place successive input regions near integer times. Their vertical half-width decreases from \(B\) by \(1/2\) per step until it reaches \(2\). The local moves can make this correction because their vertical displacement has magnitude one while the axis advances only \(\mu=1/4\). The final move will be stopped at the child mark, halfway between the last two integer times.

For a vertex set \(A\), its distance-\(R\) collar is the ambient neighborhood \(\{x\in V(G):d_G(x,A)\le R\}\), where \(d_G(x,A)=\inf_{a\in A}d_G(x,a)\) and \(\inf\varnothing=\infty\). We need each input region’s collar to contain many disjoint finite seed trials. The collar may be wide in graph distance while narrow in coordinates, because the edge-coordinate bound is chosen last.

Let \(\varepsilon\) be the permitted conditional probability of a failed lattice site. Choose it so small that \[ \sum_{l\ge1}4^l\varepsilon^{l/4}<1. \tag{50}\] The threshold \(\delta\) will test stored connection probabilities; \(\kappa\) will bound the loss at a single relay. Set \[\delta=\varepsilon/4,\qquad \kappa=\frac{\varepsilon\delta}{8(L+1)},\qquad \lambda=\kappa/4,\qquad K=\log(4/\kappa).\] The transfer proof will use the auxiliary error \(\lambda\) and trial weight \(K\). To keep accumulated endpoint errors below \(1/100\), choose \(e>0\) with \((L+3)e<1/100\). Choose the failure probability of a seed move to satisfy \[0<\eta_0\le\frac{\kappa\lambda}{4(K+1)}.\] Proposition 18, with \(e,\eta_0\), first gives a seed radius \(m\). We next choose the number \(k\) of possible contacts needed for disjoint seed trials and the graph-distance collar width \(R\). With \(D\) the maximum degree, take integers \[ \begin{split} k&>V_+(4m+2) \left\lceil K/(p_c/2)^{m+1+V_+(m)}\right\rceil,\\ R&>m+4(1-p_c)^{-Dk}/\kappa. \end{split} \tag{51}\] Now choose \(\rho>0\) so that \[ (R+m+1)\rho<e/10, \tag{52}\] and obtain the coordinates and a common \(p'\in[p_c/2,p_c)\) from Proposition 18, with edge increments at most \(\rho\). Thus \(m\) is fixed before the trial count and collar width; both are fixed before the coordinate scale and the parameter \(p'\).

At site \(v\) put \[M_v=\{0\le t-c_v^{(1)}\le\rho,\ |z-c_v^{(2)}|\le B\}.\] In a corridor \(v,x\) of sign \(\sigma\), use the successive input regions \(B_0=M_v\) and, for \(1\le i\le L\), \[ B_i=\{|t-c_v^{(1)}-i|\le(i+1)e,\ |z-c_v^{(2)}-\sigma\mu i|\le b_i\}, \qquad b_i=\max(2,B-i/2). \tag{53}\] The last target is \(B_{L+1}=M_x\).

Each distance-\(R\) collar of \(B_i\), \(0\le i\le L\), is contained in \(D_{vx}\). From every vertex \(y\) in that collar, one of the two seed moves reaches the next region with probability at least \(1-\eta_0\), using paths in \(D_{vx}\). We give the estimates, including the last crossing. Suppose first that \(0\le i<L\) and that \(y\) is in the collar of \(B_i\). A distance-\(R\) displacement changes either coordinate by at most \(R\rho\). Write \(r=z(y)-c_v^{(2)}-\sigma\mu i\) for the vertical error at the current integer time. Choose a move sign \(\tau\in\{-1,1\}\) with \(\tau r\le0\), either sign when \(r=0\). Its target error relative to the next axis point is \(r+\tau-\sigma\mu\), up to \(e/4\). With \(b_0=B\), its magnitude is at most \[\max(1+\mu,b_i+R\rho-1+\mu)+e/4\le b_{i+1}.\] The inequality follows from \(\mu=1/4\), \(b_i\ge2\), and \(R\rho+e/4<1/4\). Time error increases by at most \(R\rho+e/4<e\); at \(i=0\) the additional initial interval length \(\rho\) also fits the allowed \(2e\).

Since \(b_L=2\), a move from the collar of \(B_L\) starts below time \(T\) and ends above it. The starting vertex may be anywhere in \(B(y,m)\), so its additional time error is \(m\rho\), paid for by (52). Along the path, the first vertex at or above \(T\) has time in \([T,T+\rho]\). Its height differs from the child center by at most \(2+R\rho+C+\mu/2<B\), so it lies in \(M_x\).

For every move, while its time is between the endpoint times, its deviation from the tube axis is at most \[B+R\rho+C+\mu\bigl((L+1)e+\rho+R\rho+C\bigr)<H.\] Outside that time interval its time is within \(C+1\) of an endpoint and its height lies in that endpoint’s \(Q\). This proves containment of the paths. Collar containment follows from the same estimates without the \(C\) terms. The induced corridor therefore contains every original edge of each indicated path, as required for the seed-move probability to hold inside the corridor.

A transfer estimate in a fresh collar

The exterior-contact, independent-seed, and shell-conditioned target construction adapts Kozma and Nitzan’s Lemma 10 (Kozma and Nitzan 2024, sec. 4). We prove the version needed for fresh metric collars in the original graph, using Theorem 10.

The next argument applies to a countable product graph consisting of original edges of \(G\) and one synthetic source \(*\) wired to finitely many vertices. Edges outside the designated fresh region may have arbitrary independent probabilities, including zero and one.

Adaptive exploration with stored predictions

Stored high conditional probabilities are used in the proof of Kozma and Nitzan’s Theorem 6, particularly equations (30)–(32) (Kozma and Nitzan 2024). The adaptive block strategy has an earlier antecedent in Grimmett and Marstrand (Grimmett and Marstrand 1990, secs. 3–4). Here we use an oriented quadrant, choose one incoming corridor, reveal whole edge domains, and prove the conditional-failure boundary estimate explicitly.

Since \(p'<p_c\), Proposition 22 contradicts the assumed critical jump and completes the induction in Theorem 14.