Exact critical manifold of the five-probability inhomogeneous bow-tie lattice: full-domain proof and rigorous foundation of the negative-probability balancing construction

Ziff, Scullard, Wierman and Sedlock (2012) derived the algebraic critical manifold $B(p,r,s,t,u)=0$ for bond percolation on the bow-tie lattice with five distinct bond probabilities, by decomposing the lattice into right- and left-pointing triangular cells, tuning each cell to its self-duality condition Prob(all)=Prob(none), and recombining the two central bonds. The derivation is rigorous only when both triangles can be tuned with genuine probabilities ($p+r\le 1$ and $s+t\le 1$); the extension to the remaining admissible domain — where exactly one triangle is supercritical and an intermediate 'bond probability' is negative, reinterpreted heuristically as a sink balancing supercriticality against subcriticality — is conjectural. Because the bow-tie lattice admits no isoradial embedding, the star-triangle transport method that rigorously settled the checkerboard special case (Wu's 1979 percolation formula) does not apply. The problem is to prove or refute the full-domain validity of $B=0$, or to give the negative-parameter balancing a rigorous probabilistic or combinatorial foundation together with a criterion for which non-identical triangular arrangements admit the per-hyperedge duality argument; a certified counterexample at any admissible parameter point settles the conjecture in the negative. The problem is open.

Background

In bond percolation on an infinite connected graph, each edge (bond) is independently declared open with probability $p$ and closed otherwise; the percolation threshold $p_c$ separates a regime of only finite open clusters from one with an infinite cluster. In the inhomogeneous version, edges fall into classes with distinct probabilities $(p_1,\dots,p_n)$, and one asks for the critical manifold $f(p_1,\dots,p_n)=0$: fixing $n-1$ of the probabilities, the manifold gives the critical value of the remaining one. Only a small family of two-dimensional lattices admits exact critical manifolds, all obtained through planar duality. For a planar lattice $L$, the dual lattice $L^*$ has one vertex per face of $L$ and one dual edge crossing each edge of $L$; the classical Sykes–Essam matching argument relates the thresholds of dual pairs, and self-duality ($L\cong L^*$) can pin the threshold exactly, as for the square lattice where $p_c=1/2$.

The sharpest duality tool is the star-triangle (here: triangle-triangle) transformation: a triangular cell of bonds with edge probabilities $(a,b,c)$ can be replaced by its dual configuration with complemented probabilities, and the connectivity among the cell's three boundary vertices is preserved exactly when Prob(all)=Prob(none) — for a simple triangle, Prob(all)=$ab+ac+bc-2abc$ (all three boundary vertices mutually connected) and Prob(none)=$(1-a)(1-b)(1-c)$ (no two connected), so the condition reduces to

$$abc-a-b-c+1=0.$$

Scullard and Ziff (2008) conjectured general critical surfaces, including those for the checkerboard and bow-tie lattices, from duality and multilinearity consistency arguments, and the hypergraph formulation developed by Scullard, Ziff and collaborators expresses this as: if a lattice can be written as a self-dual 3-uniform hypergraph — a periodic arrangement of three-boundary-vertex cells ('hyperedges', each an arbitrary network of bonds with three distinguished boundary vertices) whose dual arrangement is isomorphic to itself — then imposing the per-cell condition above yields the exact critical manifold. Wierman and Ziff proved rigorous theorems of this type for self-dual planar hypergraphs, and Bollobás and Riordan proved the analogous result for self-dual polygon configurations. A second rigorous route is isoradiality: Grimmett and Manolescu proved that on graphs embedded so that the vertices of every face lie on a circle of common radius (an isoradial embedding), with bond probabilities tied to the angles subtended by the bonds through a local formula, the system is critical.

The bow-tie lattice is built from pairs of triangles joined through a shared central vertex; Wierman found its homogeneous bond threshold in 1984 via the star-triangle transform. Ziff, Scullard, Wierman and Sedlock (2012) treated the fully inhomogeneous version with five bond probabilities. Splitting the central connection into two parallel bonds, the right-pointing triangle carries probabilities $(p,r,u_1)$ and the left-pointing one $(s,t,u_2)$. Setting each triangle to its duality condition gives $u_1=(1-p-r)/(1-pr)$ and $u_2=(1-s-t)/(1-st)$; recombining the two parallel bonds, $u=u_1+u_2-u_1u_2$, yields the manifold

$$B(p,r,s,t,u)=1-u-pr-ps-rs-pt-rt-st+prs+prt+pst+rst+pru+stu-prstu=0,$$

which at $u=0$ reduces to the checkerboard polynomial $C(p,r,s,t)=1-pr-ps-rs-pt-rt-st+prs+prt+rst+pst$ of Wu's 1979 conjecture for the square lattice whose four edge probabilities repeat in a checkerboard pattern around each square.

Whenever $p+r\le 1$ and $s+t\le 1$, both $u_1$ and $u_2$ are genuine probabilities in $[0,1]$, each triangle is individually critical, and the self-dual-hypergraph theorem makes the derivation rigorous. The gap opens when, say, $p+r>1$: then $u_1<0$ — not a probability at all, but a parameter in $(-\infty,0]$ measuring that triangle's supercriticality. The authors reinterpret a negative $u_1$ as a 'sink' attached to the other central bond and conjecture that criticality is achieved by balancing supercriticality on one side against subcriticality on the other, extending $B=0$ to all regimes in which not both $p+r>1$ and $s+t>1$ (if both exceed 1 the system is supercritical for every $u$). Precisely this negative-parameter regime is needed for the checkerboard limit beyond the trivial line $p+r=s+t=1$, and there the conjecture is independently confirmed: the source paper shows the checkerboard lattice isoradially embedded, so that Wu's percolation formula follows from Grimmett and Manolescu's theorem (the $q$-state Potts extension, by contrast, remains unproven — Enting showed its conjectured $S_4$ symmetry fails for the three-state checkerboard Potts model). But the bow-tie lattice itself, in the authors' assessment, admits no isoradial embedding, so no analogous proof exists for $B=0$ at $u>0$; the same conjecture underlies the exact thresholds claimed for various asymmetric bow-tie lattices and, in Potts form, for the five-coupling frontiers of Scullard and Jacobsen — none of which are proven.

Problem Statement

Settle, with full mathematical rigor, the criticality conjecture produced by the negative-probability balancing construction of Ziff–Scullard–Wierman–Sedlock for the five-probability inhomogeneous bow-tie lattice (the split-bond bow-tie lattice: right-pointing triangles with edge probabilities $(p,r,u_1)$, left-pointing triangles with $(s,t,u_2)$, and the recombined central bond $u=u_1+u_2-u_1u_2$; equivalently, the bow-tie lattice whose five bond-probability classes are $p,r,s,t,u$):

(1) Full-domain manifold (required). Prove, or disprove, that for all $p,r,s,t,u\in[0,1]$ with not both $p+r>1$ and $s+t>1$, bond percolation on this lattice is critical exactly on

$$B(p,r,s,t,u)=1-u-pr-ps-rs-pt-rt-st+prs+prt+pst+rst+pru+stu-prstu=0.$$

The established theory covers only the region $p+r\le 1$ and $s+t\le 1$; the open regime is the one in which exactly one triangle is supercritical and the derivation passes through a negative intermediate parameter $u_1$ or $u_2$, and it also carries the exact thresholds claimed for the asymmetric bow-tie lattices generated by the same mechanism.

(2) Foundation of the balancing mechanism (optional; needed only when the answer claims generality of the method). Provide the negative-parameter ('sink') balancing step with a rigorous probabilistic or combinatorial interpretation — a proof that the algebraic balancing of a supercritical and a subcritical triangle preserves or locates criticality even though the intermediate quantity lies in $(-\infty,0]$ and is not a probability — such that the construction provably generates exact critical manifolds for the wider family of inhomogeneous lattices to which it has been applied.

(3) Admissible arrangements (optional; the method-level criterion underpinning (1)–(2)). Characterize which periodic arrangements of non-identical 3-hyperedge generators with prescribed probability assignments admit the per-hyperedge Prob(all)=Prob(none) argument (arrangements whose triangle-triangle transformation reproduces the true dual process), and which do not — the checkerboard arrangement being admissible, the 'scrambled' arrangements of the source paper's Figure 2 not, and the kagome-lattice arrangement (the lattice of corner-sharing triangles) out of reach.

These components serve one objective: placing the exact criticality of the inhomogeneous bow-tie family — the canonical non-isoradial test case of the balancing method — on rigorous footing. Any route is admissible (for example a new geometric embedding, a coupling or stochastic-domination argument, or a rigorous real-space renormalization). A complete proof of (1) by an argument that bypasses the mechanism resolves the central conjecture, in which case (2) and (3) are required only insofar as the answer claims generality of the method; a certified refutation of (1) at any admissible parameter point resolves the conjecture in the negative.

References

1. R. M. Ziff, C. R. Scullard, J. C. Wierman, M. R. A. Sedlock, The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices, J. Phys. A: Math. Theor. 45, 494005 (2012), DOI 10.1088/1751-8113/45/49/494005; arXiv:1210.6609, https://arxiv.org/abs/1210.6609
2. F. Y. Wu, Critical point of planar Potts models, J. Phys. C: Solid State Phys. 12, L645 (1979), DOI 10.1088/0022-3719/12/17/002, https://doi.org/10.1088/0022-3719/12/17/002
3. J. C. Wierman, A bond percolation critical probability determination based on the star-triangle transformation, J. Phys. A: Math. Gen. 17, 1525 (1984), DOI 10.1088/0305-4470/17/7/020, https://doi.org/10.1088/0305-4470/17/7/020
4. M. F. Sykes, J. W. Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, J. Math. Phys. 5, 1117 (1964), DOI 10.1063/1.1704215, https://doi.org/10.1063/1.1704215
5. C. R. Scullard, R. M. Ziff, Critical surfaces for general bond percolation problems, Phys. Rev. Lett. 100, 185701 (2008), DOI 10.1103/PhysRevLett.100.185701, https://doi.org/10.1103/PhysRevLett.100.185701
6. J. C. Wierman, R. M. Ziff, Self-dual planar hypergraphs and exact bond percolation thresholds, Electron. J. Combin. 18, P61 (2011), DOI 10.37236/548; arXiv:0903.3135, https://arxiv.org/abs/0903.3135
7. B. Bollobás, O. Riordan, Percolation on self-dual polygon configurations, in An Irregular Mind: Szemerédi Is 70 (Bolyai Society Mathematical Studies 21), Springer, Berlin (2010), pp. 131–217, ISBN 978-3-642-14444-8, DOI 10.1007/978-3-642-14444-8_3, https://doi.org/10.1007/978-3-642-14444-8_3 (cited as ref. [16] of the source paper)
8. G. R. Grimmett, I. Manolescu, Bond percolation on isoradial graphs: criticality and universality, Probab. Theory Relat. Fields 159, 273–327 (2014), DOI 10.1007/s00440-013-0507-y; arXiv:1204.0505, https://arxiv.org/abs/1204.0505
9. G. R. Grimmett, I. Manolescu, Inhomogeneous bond percolation on square, triangular and hexagonal lattices, Ann. Probab. 41 (4), 2990–3025 (2013), DOI 10.1214/11-AOP729; arXiv:1105.5535, https://arxiv.org/abs/1105.5535
10. C. R. Scullard, J. L. Jacobsen, Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices, J. Phys. A: Math. Theor. 46, 125006 (2013), DOI 10.1088/1751-8113/46/12/125006; arXiv:1210.7845, https://arxiv.org/abs/1210.7845
11. M. Ohzeki, Duality with real-space renormalization and its application to bond percolation, Phys. Rev. E 87, 012137 (2013), DOI 10.1103/PhysRevE.87.012137; arXiv:1211.1521, https://arxiv.org/abs/1211.1521
12. O. K. Damavandi, R. M. Ziff, Percolation on hypergraphs with four-edges, J. Phys. A: Math. Theor. 48, 405004 (2015), DOI 10.1088/1751-8113/48/40/405004; arXiv:1506.06125, https://arxiv.org/abs/1506.06125
13. N. A. M. Araújo, P. Grassberger, B. Kahng, K. J. Schrenk, R. M. Ziff, Recent advances and open challenges in percolation, Eur. Phys. J. Spec. Top. 223, 2307–2321 (2014), DOI 10.1140/epjst/e2014-02266-y; arXiv:1404.5325, https://arxiv.org/abs/1404.5325
14. C. R. Scullard, J. L. Jacobsen, R. M. Ziff, Critical percolation on the kagome hypergraph, J. Phys. A: Math. Theor. 54, 055006 (2021), DOI 10.1088/1751-8121/abcddb; arXiv:2009.01913, https://arxiv.org/abs/2009.01913
15. F. Coupette, T. Schilling, Exactly solvable percolation problems, Phys. Rev. E 105, 044108 (2022), DOI 10.1103/PhysRevE.105.044108, https://doi.org/10.1103/PhysRevE.105.044108
16. I. G. Enting, The failure of a conjectured $S_4$ symmetry in the three-state checkerboard Potts model, J. Phys. A: Math. Gen. 20, L917 (1987), DOI 10.1088/0305-4470/20/14/004, https://doi.org/10.1088/0305-4470/20/14/004