Exact critical surfaces for inhomogeneous bond percolation: the five-bond bow-tie manifold and the exactly solvable class

Scullard and Ziff's 2008 constructive method conjectured exact multilinear critical surfaces for inhomogeneous bond percolation on two-dimensional periodic lattices, with two candidate exact results — the checkerboard formula $C(p,r,s,t)=0$ and the five-bond bow-tie formula $B(p,r,s,t,u)=0$ — supported by six-digit numerics but not derivable from the known self-dual triangle-triangle construction. Subsequent work has established that the checkerboard formula has since been proved rigorously via Grimmett and Manolescu's isoradial-graph criticality theorem, while the bow-tie formula has a conditional derivation that is rigorous only on the sub-region $p+r\le1$, $s+t\le1$ and otherwise rests on a conjectural negative-probability argument; the once-intractable large-unit-cell Archimedean lattices now have reference thresholds from critical polynomials, but only numerically; and the convergence theory of the refinement schemes remains empirical. The surviving open core has two linked parts: prove or disprove $B=0$ on its full domain, and characterize when the inhomogeneous critical surface is exactly multilinear — the exactly solvable class.

Background

In bond percolation, each edge (bond) of a graph is independently declared open with probability $p$ and closed otherwise; the percolation threshold $p_c$ is the value separating the subcritical phase, in which all open clusters are finite, from the supercritical phase, in which an infinite open cluster appears. In the inhomogeneous version, different bonds carry different probabilities $(p_1,\dots,p_n)$, and the threshold generalizes to a critical surface $f(p_1,\dots,p_n)=0$: with $n-1$ probabilities fixed arbitrarily, the surface determines the critical value of the remaining one. A prominent family of testbeds is the Archimedean lattices, the eleven vertex-transitive tilings of the plane by regular polygons, denoted by the cyclic list of polygon sizes meeting at each vertex (kagome $(6,3,6,3)$, snub hexagonal $(3^4,6)$, $(4,6,12)$, and so on).

Essentially all exactly known two-dimensional thresholds follow from planar duality: the square lattice is self-dual, giving $p_c=1/2$, and the star-triangle (or $Y$–$\Delta$) transformation, which replaces a triangle of bonds by an equivalent three-legged star, extends duality arguments to further lattices. Sykes and Essam used matching-lattice counting to conjecture exact thresholds in 1964, and Wierman proved the bow-tie lattice threshold exactly in 1984 by a star-triangle substitution. A systematic extension is the triangle-triangle (cell–dual-cell) transformation of Ziff: if a lattice is built from repeated triangular cells — in the general formulation, hyperedges of a self-dual 3-uniform hypergraph, meaning arbitrary networks of bonds spanning three boundary vertices — arranged self-dually, then the system is critical exactly when, within each cell, the probability that the three corner vertices are all connected by open bonds equals the probability that none of them are connected. This criticality condition and its conclusion have been put on a rigorous footing (Chayes and Lei; Wierman and Ziff; Bollobás and Riordan).

Scullard and Ziff observed that every critical surface obtained from this condition is multilinear — of first order in each of the $n$ cell-bond probabilities, because the condition compares connection events inside a single cell — and made the converse-like working assumption that critical surfaces of general periodic lattices are multilinear even when the lattice is not of the self-dual triangular-cell type. Writing the most general multilinear function and fixing its coefficients by lattice symmetries and by agreement with all special cases that reduce to exactly solved lattices, they reproduced every known exact threshold, obtained approximations accurate to about $10^{-4}$ for the unsolved Archimedean lattices (accurate enough to be ruled out numerically as exact), and found two candidate exact surfaces that agreed with their gradient-percolation simulations to more than six digits. The first is the critical surface of the checkerboard, the square lattice with bond probabilities repeating in a $2\times2$ pattern $(p,r,s,t)$,

$C(p,r,s,t)=1-pr-ps-rs-pt-rt-st+prs+prt+pst+rst=0,$

a formula equivalent in the percolation limit to one proposed by Wu in 1979 in the Potts-model setting (bond percolation is the $q\to1$ limit of the $q$-state Potts model). The second is the critical surface of the bow-tie lattice, whose repeating unit is two triangles joined along a common central bond (a four-terminal cell, equivalently a checkerboard square with its diagonal bond present) carrying five independent bond probabilities $(p,r,s,t,u)$,

$B(p,r,s,t,u)=C(p,r,s,t)-u(1-pr-st+prst)=0,$

whose homogeneous limit $1-p-6p^2+6p^3-p^5=0$ reproduces Wierman's exact threshold $p_c=0.404518...$. Crucially, neither formula can be derived from the triangle-triangle condition, so their apparent exactness lacked any explanation, and the multilinearity assumption itself had no theoretical justification for lattices outside the self-dual class.

Two later developments frame the current status. Grimmett and Manolescu proved that the canonical bond-percolation model on any isoradial embedding of a planar graph — an embedding in which every face is inscribed in a circle of common radius, that is, a tiling by rhombi — is critical, where an edge subtending angle $\theta$ at the center of its face is open with probability $p$ given by $\tan(\theta/3)=\sqrt{3}(1-p)/(1+p)$; this isoradial criticality theorem is what now carries the proof of the checkerboard formula. Separately, Jacobsen and Scullard introduced the critical polynomial $P_B(q,v)$ of a finite basis (repeating subgraph) $B$ of a lattice, defined for the $q$-state Potts model with temperature variable $v$, whose real roots approximate the critical manifold with accuracy increasing with the size of $B$; factorization of $P_B$ under enlargement of $B$ detects exact solvability, and this method has become the standard source of reference percolation thresholds for the Archimedean lattices.

Problem Statement

The problem is the surviving open core of the question raised by Scullard and Ziff's 2008 work — determine the scope and limits of exact solvability for inhomogeneous bond-percolation critical surfaces on two-dimensional periodic lattices — after the parts resolved since 2008 (the checkerboard formula now follows from isoradial criticality [4,5]; the largest unit cells now have numerical reference thresholds [9]). A complete answer addresses the following two linked parts, which together decide, lattice by lattice, whether an exact inhomogeneous critical surface exists and what it is; each part is a necessary facet of that decision and neither substitutes for the other.

(1) Bow-tie manifold, full domain. Prove, or rigorously disprove, that $B(p,r,s,t,u)=1-u-pr-ps-rs-pt-rt-st+prs+prt+pst+rst+pru+stu-prstu=0$ is exactly the critical manifold of the bow-tie lattice with the five-probability assignment of Scullard and Ziff (2008) at every parameter point of the manifold in $[0,1]^5$: on $B=0$ the system is critical, and off the manifold it is subcritical on one side and supercritical on the other. The known triangle-cell derivation is rigorous only on the sub-region $p+r\le1$ and $s+t\le1$; on the remainder of the manifold with $u>0$, where one of the two triangular cells must be super-critical, it currently rests on a conjectural extension through negative intermediate probabilities (the sink-parameter balancing argument), whereas the checkerboard limit $u=0$ is not an open case: there $B=0$ reduces to the checkerboard surface $C(p,r,s,t)=0$, already proved by the isoradial theorem [4,5]. An acceptable solution either gives that argument a rigorous probabilistic foundation or replaces it by another method; the isoradial route that proved the checkerboard formula does not apply to the bow-tie lattice with $u>0$, since it admits no isoradial embedding. A rigorous disproof at any explicit parameter point would equally resolve this part.

(2) The exactly solvable class. Characterize by necessary and sufficient conditions — on the periodic lattice and on the assignment of probabilities to its bonds — when the inhomogeneous bond-percolation critical surface is exactly multilinear, that is, exactly given by a polynomial of first order in each bond probability of the kind constructed by the Scullard–Ziff method, thereby explaining why the linearity argument works where it does. Sufficiency is known rigorously for self-dual 3-uniform hypergraph lattices, and for the special four-terminal lattices decomposable into two fused triangular cells (bow-tie type) it is established only on the sub-region $p+r\le1$, $s+t\le1$ together with the conjectural extension described in part (1); whether such conditions are necessary — for example, whether it can be proved that the kagome lattice or the eight-probability extended checkerboard (the square lattice with eight independent bond probabilities per repeating unit cell [2]) admits no exact multilinear critical surface — is open. A complete answer must give necessary and sufficient conditions over all periodic lattices; the kagome and extended-checkerboard cases are illustrations of the necessity direction, and resolving either alone does not complete this part.

Further directions — formulating an analogue of the constructive method for site percolation and for dimension $d\ge3$, where no exact anchor results exist, and proving convergence and the scaling law of the corrections (with the effective exponents $\Delta=6,7,8$ and $\Delta=4,6,8$ measured for critical polynomials) of the approximation schemes under enlargement of the inhomogeneous region or of the critical-polynomial basis — remain open research directions without a well-defined completion boundary and are outside the scope of this task.

References

1. C. R. Scullard and R. M. Ziff, 'Critical surfaces for general bond percolation problems', Physical Review Letters 100, 185701 (2008). DOI: 10.1103/PhysRevLett.100.185701; arXiv:0805.2985. URL: https://doi.org/10.1103/PhysRevLett.100.185701
2. C. R. Scullard and R. M. Ziff, 'Critical surfaces for general inhomogeneous bond percolation problems', Journal of Statistical Mechanics (2010) P03021. DOI: 10.1088/1742-5468/2010/03/P03021; arXiv:0911.2686. URL: https://doi.org/10.1088/1742-5468/2010/03/P03021
3. R. M. Ziff, C. R. Scullard, J. C. Wierman, and M. R. A. Sedlock, 'The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices', Journal of Physics A: Mathematical and Theoretical 45, 494005 (2012). DOI: 10.1088/1751-8113/45/49/494005; arXiv:1210.6609. URL: https://doi.org/10.1088/1751-8113/45/49/494005
4. G. R. Grimmett and I. Manolescu, 'Bond percolation on isoradial graphs: criticality and universality', Probability Theory and Related Fields 159, 273–327 (2014). DOI: 10.1007/s00440-013-0507-y; arXiv:1204.0505. URL: https://doi.org/10.1007/s00440-013-0507-y
5. G. R. Grimmett and I. Manolescu, 'Universality for bond percolation in two dimensions', The Annals of Probability 41 (5), 3261–3283 (2013). DOI: 10.1214/11-AOP740; arXiv:1108.2784. URL: https://doi.org/10.1214/11-AOP740
6. C. R. Scullard and J. L. Jacobsen, 'Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices', Journal of Physics A: Mathematical and Theoretical 46, 125006 (2013). DOI: 10.1088/1751-8113/46/12/125006; arXiv:1210.7845. URL: https://doi.org/10.1088/1751-8113/46/12/125006
7. J. L. Jacobsen and C. R. Scullard, 'Critical manifold of the kagome-lattice Potts model', Journal of Physics A: Mathematical and Theoretical 45, 494003 (2012). DOI: 10.1088/1751-8113/45/49/494003; arXiv:1204.0622. URL: https://doi.org/10.1088/1751-8113/45/49/494003
8. J. L. Jacobsen, 'High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials', Journal of Physics A: Mathematical and Theoretical 47, 135001 (2014). DOI: 10.1088/1751-8113/47/13/135001; arXiv:1401.7847. URL: https://doi.org/10.1088/1751-8113/47/13/135001
9. C. R. Scullard and J. L. Jacobsen, 'Bond percolation thresholds on Archimedean lattices from critical polynomial roots', Physical Review Research 2, 012050 (2020). DOI: 10.1103/PhysRevResearch.2.012050; arXiv:1910.12376. URL: https://doi.org/10.1103/PhysRevResearch.2.012050
10. O. Khatib Damavandi and R. M. Ziff, 'Percolation on hypergraphs with four-edges', Journal of Physics A: Mathematical and Theoretical 48, 405004 (2015). DOI: 10.1088/1751-8113/48/40/405004; arXiv:1506.06125. URL: https://doi.org/10.1088/1751-8113/48/40/405004
11. F. Y. Wu, 'Critical point of planar Potts models', Journal of Physics C: Solid State Physics 12, L645–L651 (1979). DOI: 10.1088/0022-3719/12/17/002. URL: https://doi.org/10.1088/0022-3719/12/17/002
12. F. Y. Wu, 'The Potts model', Reviews of Modern Physics 54, 235–268 (1982). DOI: 10.1103/RevModPhys.54.235. URL: https://doi.org/10.1103/RevModPhys.54.235
13. J. C. Wierman, 'A bond percolation critical probability determination based on the star-triangle transformation', Journal of Physics A: Mathematical and General 17, 1525–1530 (1984). DOI: 10.1088/0305-4470/17/7/020. URL: https://doi.org/10.1088/0305-4470/17/7/020
14. L. Chayes and H. K. Lei, 'Random cluster models on the triangular lattice', Journal of Statistical Physics 122, 647–670 (2006). DOI: 10.1007/s10955-005-8078-7. URL: https://doi.org/10.1007/s10955-005-8078-7
15. J. C. Wierman and R. M. Ziff, 'Self-dual planar hypergraphs and exact bond percolation thresholds', The Electronic Journal of Combinatorics 18, P61 (2011). DOI: 10.37236/548. URL: https://doi.org/10.37236/548
16. B. Bollobás and O. Riordan, 'Percolation on self-dual polygon configurations', in An Irregular Mind (Bolyai Society Mathematical Studies 21, Springer, 2010), pp. 131–217. DOI: 10.1007/978-3-642-14444-8_3. URL: https://doi.org/10.1007/978-3-642-14444-8_3
17. M. F. Sykes and J. W. Essam, 'Exact critical percolation probabilities for site and bond problems in two dimensions', Journal of Mathematical Physics 5, 1117–1127 (1964). DOI: 10.1063/1.1704215. URL: https://doi.org/10.1063/1.1704215
18. C. R. Scullard and R. M. Ziff, 'Exact bond percolation thresholds in two dimensions', Journal of Physics A: Mathematical and General 39, 15083–15090 (2006). DOI: 10.1088/0305-4470/39/49/003. URL: https://doi.org/10.1088/0305-4470/39/49/003
19. R. M. Ziff, 'Generalized cell–dual-cell transformation and exact thresholds for percolation', Physical Review E 73, 016134 (2006). DOI: 10.1103/PhysRevE.73.016134. URL: https://doi.org/10.1103/PhysRevE.73.016134
20. M. Ohzeki, 'Duality with real-space renormalization and its application to bond percolation', Physical Review E 87, 012137 (2013). DOI: 10.1103/PhysRevE.87.012137; arXiv:1211.1521. URL: https://doi.org/10.1103/PhysRevE.87.012137
21. F. Coupette and T. Schilling, 'Exactly solvable percolation problems', Physical Review E 105, 044108 (2022). DOI: 10.1103/PhysRevE.105.044108. URL: https://doi.org/10.1103/PhysRevE.105.044108