--- title: 'Bunkbed Conjecture: Percolation Insights' url: https://www.emergentmind.com/topics/bunkbed-conjecture type: topic --- # Bunkbed Conjecture: Percolation Insights The **bunkbed conjecture** is a two-point comparison principle for independent bond percolation on the graph \(G \square K_2\), the “bunkbed” built from a finite graph \(G\) by stacking two copies of \(G\) and joining corresponding vertices by vertical edges. In its classical form, it asserts that from a vertex in one layer it is at least as likely to connect to the copy of another vertex in the same layer as to its copy in the opposite layer. Posed by Kasteleyn and long regarded as a natural geometric monotonicity statement, the conjecture generated a substantial body of partial positive results before an explicit planar counterexample showed that it is false in full generality [2410.02545]. ## 1. Definition and standard formulations Let \(G=(V,E)\) be a finite simple graph. Its bunkbed graph is commonly written either as \(G^\pm\) or \(G \square K_2\). The vertex set consists of two copies of \(V\), written \(V^-=\{v^-:v\in V\}\) and \(V^+=\{v^+:v\in V\}\). For each \(uv\in E\), there are horizontal edges \(u^-v^-\) and \(u^+v^+\), and for each \(v\in V\) there is a vertical edge \(v^-v^+\) [2204.12931]. Under independent bond percolation, each edge is open independently, with either a common parameter \(p\) or a symmetric edge-dependent family of probabilities. The basic bunkbed inequality is \[ \mathbb P(v^- \leftrightarrow w^-)\ge \mathbb P(v^- \leftrightarrow w^+), \] for all \(v,w\in V\), where \(\leftrightarrow\) denotes open-path connectivity [2204.12931]. In words, moving one endpoint “upstairs” should not increase connectivity. Several closely related formulations occur in the literature. One widely used version fixes a set \(H\subseteq V\) or \(T\subseteq V\) of posts and adds only the corresponding vertical edges, often taking those posts to be always open. In that notation the bunkbed graph is \(\operatorname{BB}(G,H)\), and the target inequality becomes \[ \mathbb P_p(v^- \leftrightarrow w^-)\ge \mathbb P_p(v^- \leftrightarrow w^+) \] for arbitrary \(v,w\in V\) [1803.07647]. A more general weighted formulation assigns a symmetric weight \(\mu\) to the bunkbed edges, requiring \[ \mathbb P_{BB(G),\mu}(x^-\sim y^+)\le \mathbb P_{BB(G),\mu}(x^-\sim y^-) \] for all \(x,y\in V(G)\) [2410.08957]. A useful hierarchy of formulations appears in the study of highly symmetric graphs. The strongest form, denoted **(B1)**, allows general edge-dependent weights \(p_G\); **(B2)** specializes to a common horizontal edge probability \(p\) with some vertical edges deterministic \(1\) or \(0\); and **(B3)** takes all horizontal and vertical edges to have the same probability \(p\). The implications \[ \text{(B1)}\Rightarrow \text{(B2)}\Rightarrow \text{(B3)} \] are explicit [2204.12931]. ## 2. Complete graphs and the first major proofs The first sustained breakthroughs occurred on complete graphs. De Buyer proved the complete-graph statement at the special parameter \(p=\tfrac12\) by a combinatorial classification of connected subgraphs of the bunkbed of \(K_n\) using triples \((x,y,z)\), where \(x\) and \(y\) count lower- and upper-layer vertices in the component and \(z\) counts “parallel vertices” whose two copies both occur. The proof introduced exact formulas for the configuration counts \(q_1(x,y,z)\) and \(q_2(x,y,z)\), together with cancellation lemmas such as \[ \sum_{i=0}^{k-z} q(k+i,k-i,z)=0, \qquad \sum_{i=0}^{k-z} q(k+i+1,k-i,z)=0, \] which force the required sign after a monotonicity analysis [1604.08439]. A subsequent proof established the complete-graph conjecture for \(p\ge \tfrac12\). In that argument, the main component containing the source is again decomposed by \((x,y,z)\), but the relevant enumerative objects are \(C_1(x,y,z)\) and \(C_2(x,y,z)\), counting configurations connecting the source to \(v\) and to \(v'\), respectively. The resulting symmetrized difference \[ C_{\mathrm{diff}}(x,y,z) = \frac{(n-2)!\,\big(x^2-2xy+y^2-x-y+2z\big)} {(x-z)!\,(y-z)!\,z!\,(n-x-y+z)!} \] has controlled sign, and two cancellation identities analogous to the earlier \(q\)-identities complete the proof [1802.04694]. The complete-graph case for **all** \(p\in[0,1]\), and for arbitrary post sets \(H\subseteq[n]\), was then settled by a symmetry-and-conditioning argument. Writing \(O\) for the open subgraph away from a distinguished target vertex \(w\), one conditions on the connected components \(c_i\) of \(O\), symmetrizes under interchange of \(w^-\) and \(w^+\), and arrives at the explicit factor \[ \left(\tilde{\mathbb P}(v^-\in c_i)-\tilde{\mathbb P}(v^+\in c_i)\right) \left(\tilde{\mathbb P}(w^+\not\sim c_i)-\tilde{\mathbb P}(w^-\not\sim c_i)\right) = \frac1n(c_i^- - c_i^+)\left(q^{c_i^+}-q^{c_i^-}\right), \] with \(q=1-p\). Since \((a-b)(q^b-q^a)\ge 0\) for \(q\in[0,1]\), each summand is nonnegative, proving \[ \mathbb P_p(v^- \leftrightarrow w^-)\ge \mathbb P_p(v^- \leftrightarrow w^+) \] on \(K_n\) [1803.07647]. These complete-graph proofs were decisive because they replaced intuitive geometric reasoning by explicit componentwise sign control. They also supplied the principal techniques later adapted to broader graph families. ## 3. Symmetric graph classes beyond \(K_n\) Richthammer extended the theory from complete graphs to several nontrivial symmetric families. The central mechanism is a pair of results for weighted complete graphs. The first handles adjacent vertices \(v,w\) under a local automorphism condition around \(w\); the second assumes that \(v\) and \(w\) have identical edge weights to every other vertex, \[ p_{vu}=p_{wu}\qquad\text{for all }u\in V\setminus\{v,w\}, \] equivalently that the transposition \(v\leftrightarrow w\) is a weighted graph automorphism [2204.12931]. In the local-symmetry theorem, a difference of target probabilities is rewritten as \[ d = 2\sum_u c_{uw}\Big(\mathbb P(u^- \leftrightarrow w^-)-\mathbb P(u^- \leftrightarrow w^+)\Big), \qquad c_{uw}=-\ln(1-p_{uw}), \] and then shown to be nonnegative by conditioning on cluster configurations away from \(w\), using automorphism symmetry and the monotonicity of \(f(x)=-\ln(1-x)\) [2204.12931]. In the equal-neighborhood theorem, reflection symmetry between the two layers, conditioning on the vertical edges at \(v\) and \(w\) and on the edge \(vw\), and a final perfect-square-type rearrangement yield nonnegativity. These two theorems imply the strong form **(B2)** for complete bipartite graphs and for complete graphs with the edges of an arbitrary complete subgraph removed. They also imply the weaker form **(B3)** for symmetric complete \(k\)-partite graphs, that is, complete multipartite graphs whose parts all have equal size [2204.12931]. The complete bipartite case was later identified as one of the “major partial (positive) results” robust enough to extend beyond percolation to the random cluster setting [2509.18788]. The importance of these results lies less in the individual classes than in the proof paradigm: global vertex-transitivity is no longer required. Local symmetry and transposition symmetry can suffice. ## 4. High-density percolation and asymptotic validity Even before the general disproof, one of the most powerful positive results was that every finite graph satisfies the bunkbed inequality when percolation is sufficiently dense. For a finite connected graph \(G=(V,E)\), there exists \(\varepsilon(G)>0\) such that for all \[ p\ge 1-\varepsilon(G), \] one has the strict inequality \[ \mathbb P_p^{\mathrm{bb}}\big((u,0)\leftrightarrow (v,0)\big) > \mathbb P_p^{\mathrm{bb}}\big((u,0)\leftrightarrow (v,1)\big) \] for every \(u,v\in V\); an explicit non-optimized choice is \[ p \ge 1-2^{-|E|/2-2}. \] A stronger conditional statement holds after conditioning on the set \(T\) of open vertical edges [2110.00282]. The proof conditions on a tripartition \(S=(S_0,S_1,S_2)\) of the horizontal edges, where \(S_0\) consists of edges whose two bunkbed copies are both closed, \(S_1\) of edges for which exactly one copy is open, and \(S_2\) of edges whose two copies are both open. After deleting \(S_0\) and contracting \(S_2\), the problem is transported to Linusson’s alternative bunkbed model on a reduced graph. The set of tripartitions is divided into three regimes: \(\mathcal S_\infty\), where a mirroring argument forces the contribution to vanish; \(\mathcal S_{\le 1}\), where one gets a uniform positive lower bound; and \(\mathcal S_{\ge 2}\), whose contribution is dominated when \(p\uparrow 1\) [2110.00282]. A later proof generalized this \(p\uparrow 1\) theorem from uniform percolation to arbitrary symmetric edge-dependent probabilities. The method is a cut expansion: if \(q(e)=1-p(e)\), one writes disconnection as a union of cut events \(E_A\), applies inclusion–exclusion, and studies the polynomial \[ f(q)=\Prob_p(u_-v_+)-\Prob_p(u_-v_-). \] The key combinatorial principle is that every negative monomial is **strictly divided** by a positive one, so for sufficiently small \(q\) the overall sign is positive on the same-layer side [2302.00031]. These high-density theorems are asymptotic rather than universal: they do not prove the conjecture for all \(p\), but they establish that any counterexample must occur away from both the sparse and dense extremes. ## 5. Forests, cactus graphs, and block-local structure A separate line of work studies graph classes for which the bunkbed inequality remains true irrespective of the general counterexample. A major structural theorem shows that the conjecture is preserved under gluing two graphs along a single vertex. If \(A\) and \(B\) both satisfy the bunkbed conjecture and \(D\) is obtained by identifying one vertex of \(A\) with one vertex of \(B\), then \(D\) also satisfies the conjecture [2410.08957]. Immediate corollaries are that the conjecture holds for forests and that any minimal counterexample must be \(2\)-connected. For forests there is also an elementary proof based on reduction to a path. If \(F\) is a forest and \(u,v\) lie in the same component, the unique path between them is relabeled \(1,2,\dots,\ell\). Conditioning on all percolation outside that path produces an effective post set \(H'\subseteq[\ell]\), reducing the problem to a path bunkbed model. In the path case, with \(m=\max H\), \[ P_{H,p}^{P_n}(1^+\leftrightarrow n^+) = P_{H,p}^{P_n}(1^+\leftrightarrow m^+)p^{n-m} = P_{H,p}^{P_n}(1^+\leftrightarrow m^-)p^{n-m} = P_{H,p}^{P_n}(1^+\leftrightarrow n^-), \] so the inequality holds with equality [2511.13589]. After the general conjecture was disproved, classification questions became central. One answer is block-local. In the strong formulation, with arbitrary edge weights \(\vec p=(p_e)_{e\in E(G)}\) and deterministic posts \(T\subseteq V(G)\), a graph satisfies the bunkbed conjecture if and only if each of its biconnected components does. The same equivalence holds for the weak version with constant edge parameter \(p\) [2506.09264]. In the two-block case the difference factorizes as \[ \mathbb P(v_0\leftrightarrow w_0)-\mathbb P(v_0\leftrightarrow w_1)=(Q-q)(R-r)\ge 0, \] which is the fundamental local product structure behind the block decomposition. This theorem yields strong positive classes. All cactus graphs satisfy the strong version. More generally, any graph whose every biconnected component is either a cycle, complete, complete bipartite, symmetric complete \(k\)-partite, or an edge difference of a complete graph and a complete subgraph satisfies the weak version [2506.09264]. Conversely, any counterexample to the strong version contains a non-trivial subdivision of the diamond graph as a minor. ## 6. Disproof, robustness, and related extensions The decisive change in status came with an explicit counterexample. There exists a connected planar graph \(G=(V,E)\) with \[ |V|=7222,\qquad |E|=14442, \] a set \(T\subseteq V\) of three transversal vertices, and vertices \(u,v\in V\) such that \[ P_{1/2}[uv] < P_{1/2}[uv']. \] This disproves the bunkbed conjecture in its original bond-percolation form [2410.02545]. The construction is built from a 3-uniform hypergraph example of Hollom by replacing each hyperedge with a planar graph gadget \(G_n\), choosing \(p=\tfrac12\) and \(n=1204\) so that a robust inequality \[ 400\, p_{a|bc}\le p_{abc}p_{a|b|c}-p_{ab|c}^2 \] holds and transfers the hypergraph imbalance to an ordinary planar graph [2410.02545]. The failure of the original conjecture is part of a broader non-robustness phenomenon. Natural generalizations to site percolation, hypergraphs, and directed graphs are all false; the site-percolation paper provides explicit counterexamples and also identifies positive cases such as paths, cycles, and wheels [2406.01790]. The directed analogue remains false even for simple acyclic directed graphs: there exists a simple acyclic directed graph with \[ \mathbb P_{1/2}(u^{-}\to v^{-}) < \mathbb P_{1/2}(u^{-}\to v^{+}), \] settling conjectures of Leander and Hollom in the negative [2506.22284]. At the same time, some partial positive results survive in broader probabilistic models. In the random cluster model, the bunkbed inequality still holds on complete graphs, on complete bipartite graphs, and for \(p\uparrow 1\) for all fixed \(q>0\), but the analogue is false for a nontrivial range \[ 0.56