Rigorous exact critical manifolds for inhomogeneous bond percolation: the five-bond bow-tie formula and the scope of multilinear critical surfaces

Scullard and Ziff's 2008 constructive method predicts critical surfaces for bond percolation on two-dimensional periodic lattices by writing the most general multilinear function of the unit-cell bond probabilities and fixing its coefficients from symmetries and exactly solved special cases. For the inhomogeneous checkerboard lattice, the resulting formula $C(p,r,s,t)=0$ - the percolation ( $q\to 1$ ) limit of Wu's 1982 checkerboard Potts conjecture - is now proven: Ziff, Scullard, Wierman and Sedlock reduced it to Grimmett and Manolescu's rigorous isoradial criticality theorem. For the five-bond inhomogeneous bow-tie lattice, however, the predicted manifold $B(p,r,s,t,u)=0$ is derived only conditionally: the generalized triangle-triangle argument is rigorous only on the sub-domain where its auxiliary intermediate probabilities are physical, the bow-tie lattice admits no isoradial embedding, and the remaining gap is the unproved 'negative probability' continuation of the self-dual triangular-lattice surface. The open problem is to prove or refute the full-domain exactness of $B=0$ .

Background

In bond percolation, every edge of a graph is independently declared open with probability $p$ and closed otherwise, and one asks when an infinite connected cluster of open edges appears. The threshold $p_c$ at which this happens depends on the lattice, and computing it is a classical unsolved problem in general: outside two dimensions no exact thresholds are known at all, and even in two dimensions exact results exist only for special lattice classes, all rooted in planar duality (the Sykes-Essam relation between a planar lattice and its planar dual, whose bonds are open with complementary probability $1-p$ ; self-duality of the square lattice underlies the exact threshold $p_c=1/2$ ). In the inhomogeneous version, different bonds carry different probabilities $(p_1,\dots,p_n)$ on a periodic unit cell, and the threshold generalizes to a critical surface or critical manifold $f(p_1,\dots,p_n)=0$ : with $n-1$ probabilities fixed arbitrarily, the root gives the critical value of the remaining one. The star-triangle transformation - replacing a triangle of bonds by a three-bond star at special probability values that leave the connection probabilities between the outer vertices unchanged - is the engine behind the known exact surfaces; a standard example is the honeycomb surface $H(p,r,s)=prs-rp-rs-ps+1=0$ . Ziff and Scullard generalized this to a triangle-triangle (self-dual 3-uniform hypergraph) construction: any network of bonds inside three boundary terminals can serve as a cell, and if the cells are arranged self-dually the criticality condition is that the probability all three terminals are connected equals the probability none are. Because this condition involves only events on one cell, every critical function it produces is multilinear - first order in each bond probability - so the homogeneous threshold is the root of a polynomial of degree at most the number of cell bonds. Scullard and Ziff (2008) then conjectured, as a constructive ansatz, that multilinearity holds even for lattices outside this solvable class: write the most general multilinear function (with $2^n$ unknown coefficients for $n$ cell bonds) and fix the coefficients by lattice symmetries, by invariance under repartitioning the lattice into different unit cells, and by reduction to exactly solved special cases. The method reproduces every known exact result and gives four-significant-figure approximations for the unsolved Archimedean lattices (the eleven vertex-transitive tilings of the plane, such as kagome and $(4,8^2)$ ; the notation $(4,8^2)$ lists the sizes of the polygons meeting at each vertex, here one square and two octagons), which the authors themselves showed cannot be exact - the single-cell multilinear predictions are ruled out by numerical thresholds - so multilinearity is not universal. The ansatz also admits refinement: extending the inhomogeneous region over several unit cells, with an independent probability on each bond, adds unknowns but yields successively better approximations, empirically, as the region is enlarged. Two predictions, however, agreed with gradient-percolation simulations (simulations in which the occupation probability varies slowly with position, so that the percolation frontier appears directly as a sharp boundary whose location can be measured very precisely) to six or more digits and were flagged as possibly exact. One is the checkerboard formula $C(p,r,s,t)=1-pr-ps-rs-pt-rt-st+prs+prt+pst+rst=0$ for the square lattice whose four bond probabilities alternate in a $2\times 2$ pattern; this is exactly the $q\to 1$ percolation limit of a critical surface conjectured by Wu in 1982 for the checkerboard $q$ -state Potts model (a lattice spin model with $q$ possible states per site in which equal neighboring states are energetically favored; its $q\to 1$ limit is bond percolation), a conjecture known to fail at $q=3$ , to hold at $q=2$ (Ising), and untested at $q=1$ before this work. The other is the five-bond inhomogeneous bow-tie formula $B(p,r,s,t,u)=1-u-pr-ps-rs-pt-rt-st+prs+prt+pst+rst+pru+stu-prstu=0$ for the bow-tie lattice (a square lattice with one diagonal per cell), whose homogeneous specialization $1-p-6p^2+6p^3-p^5=0$ reproduces Wierman's rigorous threshold $p_c=0.404518\ldots$ obtained by star-triangle substitution. Neither formula follows from the self-dual cell condition. The state of the art on the rigorous side is the isoradial program: an isoradial embedding is a planar drawing in which every face is a cyclic polygon of circumradius one with the circumcenter inside the face, equivalent to a tiling of the plane by rhombi; assigning each bond the probability fixed by the angle it subtends defines canonical isoradial percolation, and Grimmett and Manolescu proved rigorously that canonical bond percolation on such embeddings (with mild regularity conditions, including all periodic cases) is critical, confirming a long-standing conjecture in the isoradial percolation program. On the numerical side, the multilinear ansatz has been superseded by the critical-polynomial method, which associates to a finite basis of $n\times n$ unit cells a graph polynomial whose root estimates $p_c$ with rapidly improving accuracy, reproduces all exact solutions through factorization, and has produced reference thresholds for all eleven Archimedean lattices.

Problem Statement

Establish the remaining rigorous content of the constructive theory of exact percolation critical surfaces by resolving its flagship open item: prove or refute the conjecture of Ziff, Scullard, Wierman and Sedlock (2012) that $B(p,r,s,t,u)=1-u-pr-ps-rs-pt-rt-st+prs+prt+pst+rst+pru+stu-prstu=0$ is the exact critical manifold of bond percolation on the bow-tie lattice (a square lattice with one diagonal per unit cell) with the five independent bond probabilities of its unit cell, over the entire parameter domain $(p,r,s,t,u)\in[0,1]^5$ . In the source's convention, $u$ is the diagonal bond of the cell and $p,r,s,t$ are the four remaining bonds, entering as the pairs $\{p,r\}$ and $\{s,t\}$ carried by the right- and left-pointing triangles of the triangle-triangle construction; indeed $B = C(p,r,s,t) - u(1-pr)(1-st)$ , so a relabeling that does not preserve these pairs yields a different conjectured manifold. The criterion is the operational one used in the source: with four of the five bond probabilities fixed arbitrarily in $[0,1]$ , the root of $B=0$ in the remaining probability gives the critical value separating absence of an infinite open cluster from its presence as that probability varies. Where an assignment degenerates - for instance where the coefficient $(1-pr)(1-st)$ of $u$ vanishes, so that $B$ reduces to the proven checkerboard polynomial $C(p,r,s,t)$ and no longer determines $u$ - the operational criterion is understood to apply to the probabilities that $B=0$ does determine, without any further restriction on the domain $[0,1]^5$ . The conjecture is read at the assignments at which the free probability admits a genuine transition separating a subcritical regime (no infinite open cluster) from a supercritical one; at assignments admitting no such transition - in particular assignments with a fixed probability equal to $0$ or $1$ for which it fails to occur - the conjecture asserts nothing, and such an assignment neither confirms nor refutes it. A refutation consists in exhibiting a concrete parameter assignment, admitting such a genuine transition, at which $B=0$ demonstrably fails to give the critical value.

The present derivation is rigorous only on the sub-domain where the auxiliary triangle-triangle construction keeps its intermediate split-bond probabilities $u_1=(1-p-r)/(1-pr)$ and $u_2=(1-s-t)/(1-st)$ in $[0,1]$ , i.e. where $p+r\le 1$ and $s+t\le 1$ ; a complete answer must either supply a full rigorous argument for the entire domain (for instance by giving the negative-probability continuation of the self-dual triangular surface a precise and provable meaning, or by an independent method), or exhibit such a concrete failing assignment. Known exact special cases that any answer must respect include the honeycomb surface $H(p,r,s)=prs-rp-rs-ps+1=0$ , Wierman's rigorous homogeneous bow-tie threshold as the root of $1-p-6p^2+6p^3-p^5=0$ , and the proven checkerboard surface $C(p,r,s,t)=0$ recovered at $u=0$ . The source work notes that the same constructive procedure lacks exact anchor solutions in site percolation and in dimension $d\ge 3$ ; those extensions have no sharp completion boundary and lie outside the bounded problem stated here.

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