Smirnov's 2001 proof of conformal invariance and Cardy's formula for critical site percolation on the triangular lattice, together with the identification of single-interface scaling limits with chordal SLE6_6, left open three directions: a detailed construction of the full continuum percolation configuration (all nested cluster boundaries), a complete multi-curve description of the scaling limit, and the extension to lattices other than the triangular site model. The first two directions were closed for the triangular lattice by Camia and Newman (2004-2008), who constructed the full scaling limit as a unique, conformally invariant ensemble of nested loops identified with CLE6_6. The surviving open core is the third direction: establish the same conformally invariant scaling limit for critical percolation on other planar lattices, canonically bond percolation on the square lattice at pc=1/2p_c = 1/2. This requires proving convergence of crossing probabilities to Cardy's hypergeometric formula in arbitrary domains, convergence of exploration paths to SLE6_6, and existence, uniqueness, and conformal invariance of the full loop-ensemble limit, without the triangular lattice's discrete-holomorphic color-switching mechanism, for which no substitute is known.

Background

Percolation models a random porous medium: in Bernoulli bond percolation on a planar lattice each edge is independently declared open with probability pp, and in site percolation each vertex is. As pp crosses a lattice-dependent critical value pcp_c an infinite open cluster appears; the model at p=pcp = p_c is called critical. For bond percolation on the square lattice Z2\mathbb{Z}^2 one has pc=1/2p_c = 1/2 (Kesten), and for site percolation on the triangular lattice also pc=1/2p_c = 1/2. A central observable is the crossing probability: the probability, in a bounded simply connected domain with four marked boundary points, that an open cluster connects one designated boundary arc to the opposite arc. Physics arguments from two-dimensional conformal field theory (the quantum-field-theoretic description of critical two-dimensional systems, in which observables transform covariantly under conformal maps) predict that at criticality, as the lattice mesh δ→0\delta \to 0, these probabilities become conformally invariant — unchanged under conformal (angle-preserving) maps of the domain — and equal an explicit hypergeometric function of the conformal modulus of the marked domain, that is, the aspect ratio of the rectangle onto which conformal maps send the marked domain; this modulus is a complete invariant of marked domains up to conformal equivalence. This prediction is Cardy's formula. Numerical evidence for conformal invariance goes back to Langlands, Pouliot and Saint-Aubin (the conjecture is attributed by them to Aizenman), and Cardy gave the predicted formula in 1992. The corresponding prediction for interfaces is that, in the scaling limit, percolation cluster boundaries become random conformally invariant curves.

The stochastic Loewner evolution SLEκ_\kappa (Schramm, 2000) is the one-parameter family of random curves obtained from Loewner's classical description of a growing planar hull driven by one-dimensional Brownian motion with diffusivity parameter κ\kappa; SLE curves are conformally invariant and are the proved or conjectured scaling limits of critical two-dimensional interfaces. The chordal variant, which grows from one prescribed boundary point to another, is the one relevant here. The value κ=6\kappa = 6 has the locality property (the curve interacts with independent percolation-like input in a Markovian way) and is the universality class of percolation. The percolation exploration path is the interface between open and closed clusters produced by revealing colors progressively along a boundary arc; it is defined on any planar lattice. The conformal loop ensemble CLE6_6 is the corresponding loop-level object: a conformally invariant random collection of nested, non-simple loops that describes all cluster boundaries at once (the loop analog of SLE).

Smirnov (2001) proved, for critical site percolation on the triangular lattice, that certain lattice observables — probabilities that an interior point is separated from a boundary arc by a cluster path — are asymptotically discrete-harmonic, via an exact color-switching identity that couples three such observables into a 'Cauchy–Riemann triple' (a lattice form of the Cauchy–Riemann equations of complex analysis) reflecting the 2π/32\pi/3 rotational symmetry of the lattice. From this he obtained conformal invariance and Cardy's formula for crossing probabilities (Theorem 1 and Corollaries 2–4), convergence of the outer boundary of a cluster and of the exploration path to chordal SLE6_6 (Corollaries 5 and 7), and stated — with only a proof sketch — that the law of the collection of all cluster external perimeters, a nested family of oriented loops viewed as Hölder-continuous curves up to reparameterization, converges to a unique conformally invariant limit (Theorem 4). The proof rests on the lattice-specific holomorphic identity plus two lattice-independent inputs: Russo–Seymour–Welsh (RSW) bounds, which give uniform upper and lower bounds on crossing probabilities of annuli of fixed aspect ratio, and Aizenman–Burchard precompactness, which gives tightness of laws of non-self-crossing curves with uniformly bounded Hölder norms.

For the triangular site model this program was then completed by Camia and Newman (2004–2008): they constructed the full scaling limit, the continuum ensemble of all nested loops inherited from all cluster boundaries, proved its existence, uniqueness, and conformal invariance, gave a complete proof that the exploration path converges to chordal SLE6_6, and identified the loop ensemble with CLE6_6. The passage from the crossing-probability statement to the interface and full-scaling-limit statements proceeds by precompactness together with determination of the limit law by crossing events. Later refinements — Hongler–Smirnov (2010/2011, a conformally invariant observable given by the expected number of clusters separating two pairs of points), Khristoforov–Smirnov (2021, an 'ultimate' proof of Cardy's formula via the O(1) loop model, a lattice model of non-intersecting loops carrying weight 1 per loop), Khristoforov–Skopenkov–Smirnov (2023/2025), and Camia (2024, conformal covariance of multi-point connection probabilities) — all remain on the hexagonal/triangular lattice.

What has resisted all efforts is the passage to other lattices. On the square lattice the RSW input is available (by self-duality), but the discrete-holomorphic color-switching identity is a rigid consequence of the triangular lattice's three-fold symmetry and has no known analogue on Z2\mathbb{Z}^2; the parafermionic observables — complex-valued lattice observables introduced in the random-cluster (Fortuin–Kasteleyn) representation, which generalizes percolation — satisfy only partial discrete-holomorphicity relations at the percolation point and do not close into a proof. Consequently, conformal invariance — even of a single crossing probability in a single non-symmetric domain — is unproved for critical bond percolation on Z2\mathbb{Z}^2 and for site percolation on Z2\mathbb{Z}^2 (whose exact pcp_c is itself unknown).

Problem Statement

Prove conformal invariance and identify the scaling limit of a critical Bernoulli percolation model on a standard planar lattice outside the hexagonal/triangular family — that is, neither the hexagonal/triangular lattice itself nor a model that reduces to it by an exact lattice transformation mapping it to the hexagonal/triangular site model while preserving crossing probabilities (such as the bond-triangular-type models or the 3-12 lattice, for which Cardy's formula or conformal invariance is already known): it suffices to establish (a)-(c) below for any one such lattice, the flagship case being critical bond percolation on the square lattice Z2\mathbb{Z}^2 at p=1/2p = 1/2; site percolation on Z2\mathbb{Z}^2 or other genuinely independent periodic planar lattices are equally admissible targets. Concretely, as the lattice mesh δ→0\delta \to 0: (a) Cardy's formula and conformal invariance of crossing probabilities: in every simply connected domain with four marked boundary points, the probability of an open crossing between two designated boundary arcs converges to Cardy's hypergeometric function of the conformal modulus of the marked domain, and is therefore conformally invariant; (b) interface convergence: the exploration path between two boundary points converges in distribution to chordal SLE6_6; and (c) full scaling limit: the collection of all cluster external perimeters — the nested family of oriented, non-self-crossing loops representing the entire continuum percolation configuration — converges to a unique, conformally invariant limit (expected, though not required for completion, to coincide with the CLE6_6-type loop ensemble constructed for the triangular lattice). These three parts form one linked program with a single objective, rigorous universality of the conformally invariant percolation scaling limit: part (a) is the crux, and parts (b) and (c) are expected to follow from (a). Establish (a) in full generality — arbitrary domains, not special geometries — together with (b) and (c), or a rigorous proof that (b) and (c) follow from (a) for the chosen lattice. The result must hold in arbitrary domains, not merely special geometries, and unconditionally — without appeal to unproved inputs such as rotational invariance of the scaling limit. A rigorous disproof — a demonstration that the conformally invariant Cardy limit fails for the chosen lattice — would equally settle the question.

References

  1. Stanislav Smirnov, 'Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits', Comptes Rendus de l'Académie des Sciences – Series I – Mathematics 333 (2001) 239–244, DOI 10.1016/s0764-4442(01)01991-7, https://doi.org/10.1016/s0764-4442(01)01991-7 (full preprint text: arXiv:0909.4499, https://arxiv.org/abs/0909.4499)
  2. John L. Cardy, 'Critical percolation in finite geometries', Journal of Physics A: Mathematical and General 25 (1992) L201–L206, DOI 10.1088/0305-4470/25/4/009, https://doi.org/10.1088/0305-4470/25/4/009
  3. Robert Langlands, Philippe Pouliot and Yvan Saint-Aubin, 'Conformal invariance in two-dimensional percolation', Bulletin of the American Mathematical Society 30 (1994) 1–61, DOI 10.1090/s0273-0979-1994-00456-2, https://doi.org/10.1090/s0273-0979-1994-00456-2
  4. Oded Schramm, 'Scaling limits of loop-erased random walks and uniform spanning trees', Israel Journal of Mathematics 118 (2000) 221–288, DOI 10.1007/bf02803524, https://doi.org/10.1007/bf02803524
  5. Michael Aizenman and Almut Burchard, 'Hölder regularity and dimension bounds for random curves', Duke Mathematical Journal 99 (1999) 419–453, DOI 10.1215/s0012-7094-99-09914-3, https://doi.org/10.1215/s0012-7094-99-09914-3
  6. Harry Kesten, 'The critical probability of bond percolation on the square lattice equals 1/2', Communications in Mathematical Physics 74 (1980) 41–59, DOI 10.1007/bf01197577, https://doi.org/10.1007/bf01197577
  7. Federico Camia and Charles M. Newman, 'Continuum Nonsimple Loops and 2D Critical Percolation', Journal of Statistical Physics 116 (2004) 157–173, DOI 10.1023/b:joss.0000037221.31328.75, https://doi.org/10.1023/b:joss.0000037221.31328.75
  8. Federico Camia and Charles M. Newman, 'Two-Dimensional Critical Percolation: The Full Scaling Limit', Communications in Mathematical Physics 268 (2006) 1–38, DOI 10.1007/s00220-006-0086-1, https://doi.org/10.1007/s00220-006-0086-1 (preprint arXiv:math/0504036, https://arxiv.org/abs/math/0504036)
  9. Federico Camia and Charles M. Newman, 'Critical percolation exploration path and SLE6: a proof of convergence', Probability Theory and Related Fields 139 (2007) 473–519, DOI 10.1007/s00440-006-0049-7, https://doi.org/10.1007/s00440-006-0049-7
  10. Federico Camia and Charles M. Newman, 'SLE6_6 and CLE6_6 from critical percolation', in Probability, Geometry and Integrable Systems, Cambridge University Press (2008) 103–130, DOI 10.1017/9781009701242.006, https://doi.org/10.1017/9781009701242.006
  11. Clément Hongler and Stanislav Smirnov, 'Critical percolation: the expected number of clusters in a rectangle', Probability Theory and Related Fields 151 (2011) 735–756, DOI 10.1007/s00440-010-0313-8, https://doi.org/10.1007/s00440-010-0313-8 (online first 2010; preprint arXiv:0909.4490, https://arxiv.org/abs/0909.4490)
  12. Mikhail Khristoforov and Stanislav Smirnov, 'Percolation and O(1) loop model', arXiv:2111.15612 (2021, preprint), https://arxiv.org/abs/2111.15612
  13. Mikhail Khristoforov, Mikhail Skopenkov and Stanislav Smirnov, 'A generalization of Cardy's and Schramm's formulae', Communications in Mathematical Physics 406 (2025), no. 5, DOI 10.1007/s00220-025-05255-z, https://doi.org/10.1007/s00220-025-05255-z (preprint arXiv:2305.12368, https://arxiv.org/abs/2305.12368)
  14. Federico Camia, 'Conformal covariance of connection probabilities and fields in 2D critical percolation', Communications on Pure and Applied Mathematics 77 (2024) 2138–2176, DOI 10.1002/cpa.22171, https://doi.org/10.1002/cpa.22171 (online first 2023; preprint arXiv:2203.08167, https://arxiv.org/abs/2203.08167)
  15. Hugo Duminil-Copin, 'Sixty years of percolation', arXiv:1712.04651 (2017), https://arxiv.org/abs/1712.04651
  16. L. Chayes and H. K. Lei, 'Cardy's formula for certain models of the bond-triangular type', Reviews in Mathematical Physics 19 (2007) 511–565, DOI 10.1142/s0129055x0700305x, https://doi.org/10.1142/s0129055x0700305x
  17. Jonathan Tsai, S. C. P. Yam and Wang Zhou, 'Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice', arXiv:1112.2017 (2011, unpublished preprint), https://arxiv.org/abs/1112.2017
  18. Wang Zhou, 'SLE6SLE_6 and 2-d critical bond percolation on the square lattice', arXiv:2409.03235 (2024, unpublished preprint), https://arxiv.org/abs/2409.03235
  19. Junyu Mou and Hao Wu, 'Ising model and percolation: from hexagonal lattice to 3-12 lattice', arXiv:2606.00945 (2026, preprint), https://arxiv.org/abs/2606.00945