Benjamini's conjecture: the limiting shape of intrinsic-metric balls in the supercritical percolation infinite cluster becomes round as p approaches p_c

For supercritical site percolation on the triangular lattice $\mathbb{T}$ ($p>p_c=1/2$), condition the origin to belong to the a.s. unique infinite open cluster and consider the balls of radius $n$ centered at the origin in the chemical (graph) distance of that cluster. The conjecture of Itai Benjamini — posed in his ICM 2010 article and Erdős Centennial chapter and restated as an open question in Duminil-Copin [1] — asserts that these balls possess a deterministic limiting shape $U_p$ and that $U_p$, suitably normalized, becomes round, i.e. tends to a Euclidean disk, as $p\searrow p_c$. The existence of the limit shape $U_p$ is known by the shape theorem of Garet–Marchand in its triangular-lattice adaptation, while the roundness of the normalized shape as $p\searrow p_c$ remains open.

Background

Site percolation on the triangular lattice $\mathbb{T}$ assigns to each vertex (site) the state open with probability $p$ and closed with probability $1-p$, independently across sites. The critical point is $p_c=1/2$; at and near criticality one has Smirnov's conformal invariance (the theorem that crossing probabilities of critical site percolation on $\mathbb{T}$ become conformally invariant in the scaling limit) and RSW bounds (the Russo–Seymour–Welsh uniform lower bounds on rectangle-crossing probabilities). For every $p>p_c$ there exists almost surely a unique infinite open cluster, denoted $\mathcal{C}_p^\infty$. The chemical distance (also called graph distance or intrinsic distance) $d_{\mathcal{C}}(x,y)$ is the minimum number of edges of any open path joining $x$ to $y$ within the cluster; it is infinite if $x,y$ are not connected by open sites. Antal and Pisztora proved that, for all dimensions $d\geq 2$, the chemical distance between two connected points is at most a constant (depending only on $p$ and $d$) times their Euclidean distance, with exponential tails. Garet and Marchand then proved a full asymptotic shape theorem on the infinite Bernoulli cluster: there exists a deterministic function $\mu^{\mathrm{chem}}_p(z)$, the chemical-distance time constant, such that, conditional on $0$ and $ny$ being connected, $d_{\mathcal{C}}(0,ny)/n\to\mu^{\mathrm{chem}}_p(y)$ almost surely for each fixed $y\in\mathbb{Z}^d\setminus\{0\}$; $\mu^{\mathrm{chem}}_p$ is symmetric, homogeneous and subadditive, hence extends to a norm for $p>p_c$, and the chemical-distance balls of radius $n$, rescaled by $1/n$, converge almost surely in Hausdorff distance to the compact convex set $U_p=\{z:\mu^{\mathrm{chem}}_p(z)\leq 1\}$. The published statements are for bond percolation on $\mathbb{Z}^d$; the renormalization arguments adapt to site percolation on the triangular lattice, as Yao explicitly remarks for the Antal–Pisztora bounds, so the content of the conjecture is its second clause. Subsequent work refined the regularity of $p\mapsto\mu^{\mathrm{chem}}_p$ — continuity on $(p_c,1]$ (Garet–Marchand–Procaccia–Théret), an $\epsilon|\log\epsilon|$ modulus on every interval $[p_0,1]$ with $p_0>p_c$ (Dembin), Lipschitz continuity on every such interval (Cerf–Dembin, using a delicate bypass construction), sublinear variance of the point-to-point chemical distance (Dembin), and an upper-tail large-deviation rate function (Dembin–Nakajima) — but all of these hold only away from criticality and none controls the near-critical window $p\searrow p_c$.

A parallel line of work concerns the asymptotic shape of large finite clusters, an extrinsic question. In the subcritical regime, the inverse correlation length $\tau_p(u)=\lim_{n\to\infty}-\frac{1}{n}\log\mathbb{P}_p(0\leftrightarrow\lfloor nu\rfloor)$ defines a surface tension, and the associated Wulff crystal — the compact convex set whose support function is $\tau_p$ — governs the law of large conditioned clusters. Duminil-Copin [1] proves that on the triangular lattice $\tau_p(u)/\tau_p(|u|)\to 1$ uniformly over directions as $p\nearrow p_c$, so that the volume-normalized Wulff crystal converges to a Euclidean disk; the proof inputs the rotational invariance of the near-critical scaling limit of percolation constructed by Garban, Pete and Schramm (a scaling limit, as $p\uparrow p_c$ with the mesh tending to 0, taken in the quad-crossing topology — Schramm's topology in which a percolation configuration is specified by which quadrilaterals are crossed by open paths — with the lattice scale replaced by a continuum). The supercritical extrinsic analog follows by self-duality/matching duality on the triangular lattice (the lattice symmetry exchanging open and closed sites). Two further roundness results of the same extrinsic type are due to Chang-Long Yao. The first concerns near-critical Bernoulli first-passage percolation (the model in which each site carries a random passage time and the travel time between two points is the infimum of total times over connecting paths; in Yao's theorems one color of sites is charged 1 and the other 0): its limit shape, rescaled by the correlation length $L(p)$ (the characteristic scale, diverging as $p\downarrow p_c$, below which the configuration at $p$ still looks critical), converges to a disk as $p\uparrow p_c$. The second is the Wulff crystal for anchored isoperimetry of the supercritical infinite cluster, which converges to a disk as $p\downarrow p_c$, settling, for site percolation on the triangular lattice, the site version of a conjecture of Biskup–Louidor–Procaccia–Rosenthal for bond percolation. Manolescu and Mohanarangan recently extended the Wulff roundness to FK-percolation (the Fortuin–Kasteleyn random-cluster model, a correlated generalization of Bernoulli percolation with cluster weight $q$) with $q>4$: they prove the correlation length becomes isotropic as $q$ approaches its self-dual value $4$.

A crucial technical distinction separates all of these results from Benjamini's conjecture. The first-passage-percolation time constant $\mu^{\mathrm{fpp}}_p$ in Yao's roundness theorems charges passage time 1 for open sites and 0 for closed sites (or vice versa), so paths may travel through closed sites free of charge; Yao proves this time constant equals the isoperimetric boundary norm $\beta_p$ of the cluster (Theorem 1.5 of the 2023 paper establishes $L(p)\mu^{\mathrm{fpp}}_p(z)\to\nu$ uniformly in the direction $z$, for a constant $\nu$ not depending on $p$ or $z$). This is a boundary-type norm, governed by crossing probabilities and surface tension, i.e. by objects accessible to the conformally invariant scaling limit. The chemical-distance norm $\mu^{\mathrm{chem}}_p$ instead forbids closed sites entirely and accumulates the cost of detouring around holes of the cluster on all scales; it has no known representation in terms of connection probabilities, and it is not identified with any norm for which roundness is known. In the words of the literature, metric properties do not follow from conformal geometry.

Away from criticality the intrinsic shape is known to be genuinely anisotropic: as $p\to 1$, $\mu^{\mathrm{chem}}_p$ converges to the $\ell_1$ norm, and Basdevant, Enriquez, Gerin and Gouéré proved that large balls are asymptotically diamond-shaped, with boundary arcs near the axes described by pieces of parabolas (via a correspondence with the stationary measure of the totally asymmetric simple exclusion process); within the oriented-percolation cone (the set of directions in which oriented, say northeast-only, percolation is itself supercritical) the time constant equals the $\ell_1$ norm exactly. Roundness is thus specifically a near-critical phenomenon, the conjecture being that the intrinsic geometry of the supercritical cluster inherits the rotational invariance of the critical point. At criticality itself the intrinsic geometry is nontrivial: the chemical distance between boundary points of critical clusters grows superlinearly with an exponent strictly larger than $1$ (Damron–Hanson–Sosoe), and the intrinsic metric and random walk on critical clusters were recently shown to admit a rotationally invariant scaling limit, identified with the geodesic metric of the CLE6 gasket and with CLE6 Brownian motion (Đanković–Markering–Miller–Yuan), where CLE6 denotes the conformal loop ensemble with parameter $\kappa=6$ — the universal loop ensemble describing scaling limits of critical percolation interfaces — and its gasket is the set left uncovered by its outermost loops. These critical results do not control the supercritical near-critical window, in which even the blow-up rate of $\mu^{\mathrm{chem}}_p$ as $p\searrow p_c$ appears to be open.

Problem Statement

Work with site percolation on the triangular lattice $\mathbb{T}$ at parameter $p\in(p_c,1)$, where $p_c=1/2$, and condition on the event that the origin belongs to the a.s. unique infinite open cluster $\mathcal{C}_p^\infty$. Let $B_p(0,n)=\{x\in\mathcal{C}_p^\infty:d_{\mathcal{C}_p^\infty}(0,x)\leq n\}$ be the ball of radius $n$ centered at the origin in the chemical (graph) distance on the infinite cluster. The existence clause is established background: almost surely under the conditional measure, $n^{-1}B_p(0,n)$ converges in Hausdorff distance (the standard distance between compact sets: the greatest distance from a point of one set to the other) to a deterministic compact convex set $U_p$ (the unit ball of the chemical-distance time constant), by the shape theorem of Garet–Marchand in its triangular-lattice adaptation. Prove the open core: as $p\searrow p_c$, the normalized shape becomes round — for instance, $U_p$ rescaled to have area $\pi$ converges in Hausdorff distance to the Euclidean unit disk, equivalently $\mu^{\mathrm{chem}}_p(z)/\mu^{\mathrm{chem}}_p(|z|)\to 1$ uniformly over directions $z$ on the unit circle as $p\searrow p_c$, where $\mu^{\mathrm{chem}}_p$ is the chemical-distance time constant. The claim is a full-limit claim: it must hold for all $p$ sufficiently close to $p_c$ and all directions, not merely along a sequence of parameters or a restricted set of directions. The statement concerns the intrinsic (chemical) metric in which closed sites are impassable — this is what distinguishes it from the roundness theorems for the first-passage and isoperimetric norms, in which one color of sites is traversed free of charge. A refutation would be a rigorous proof that the normalized $U_p$ stays bounded away from the disk as $p\searrow p_c$, e.g. a flat edge of non-vanishing relative size or a directional ratio bounded away from 1 along a sequence $p_k\searrow p_c$.

References

1. Hugo Duminil-Copin (2013), 'Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattice', Electronic Communications in Probability 18. DOI: 10.1214/ECP.V18-3163; arXiv:1305.6034. https://arxiv.org/abs/1305.6034 (the published ECP record carries the title with the typographical truncation 'lattic', while the arXiv version ends 'lattice')
2. Itai Benjamini, 'Random Planar Metrics', in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), Hyderabad, vol. IV, pp. 2177–2187, Hindustan Book Agency (congress held 2010; the volume was issued in June 2011). DOI: 10.1142/9789814324359_0140. https://doi.org/10.1142/9789814324359_0140
3. Itai Benjamini (2013), 'Euclidean vs. Graph Metric', in Erdős Centennial, Bolyai Society Mathematical Studies 25, pp. 35–57. DOI: 10.1007/978-3-642-39286-3_2. https://doi.org/10.1007/978-3-642-39286-3_2
4. Peter Antal and Ágoston Pisztora (1996), 'On the chemical distance for supercritical Bernoulli percolation', The Annals of Probability 24(2), 1036–1048. DOI: 10.1214/aop/1039639377. https://doi.org/10.1214/aop/1039639377
5. Olivier Garet and Régine Marchand (2004), 'Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster', ESAIM: Probability and Statistics 8, 169–199. DOI: 10.1051/ps:2004009. https://doi.org/10.1051/ps:2004009 (the preprint arXiv:math/0304144, https://arxiv.org/abs/math/0304144, is this paper under its earlier title 'Asymptotic shape for the chemical distance and first-passage percolation in random environment')
6. Olivier Garet and Régine Marchand (2007), 'Large deviations for the chemical distance in supercritical Bernoulli percolation', The Annals of Probability 35(3), 833–866. DOI: 10.1214/009117906000000881. https://doi.org/10.1214/009117906000000881
7. Olivier Garet, Régine Marchand, Eviatar B. Procaccia and Marie Théret (2017), 'Continuity of the time and isoperimetric constants in supercritical percolation', Electronic Journal of Probability 22. DOI: 10.1214/17-ejp90; arXiv:1512.00742. https://arxiv.org/abs/1512.00742
8. Barbara Dembin (2021), 'Regularity of the time constant for a supercritical Bernoulli percolation', ESAIM: Probability and Statistics 25, 109–132. DOI: 10.1051/ps/2021005; arXiv:1803.03141. https://arxiv.org/abs/1803.03141
9. Raphaël Cerf and Barbara Dembin (2022), 'The time constant for Bernoulli percolation is Lipschitz continuous strictly above p_c', The Annals of Probability 50(5). DOI: 10.1214/22-aop1565; arXiv:2101.11858. https://arxiv.org/abs/2101.11858 (the published title renders 'p_c' as 'pc')
10. Barbara Dembin (2024), 'The variance of the graph distance in the infinite cluster of percolation is sublinear', ALEA, Lat. Am. J. Probab. Math. Stat. 21. DOI: 10.30757/alea.v21-13; arXiv:2203.01083. https://arxiv.org/abs/2203.01083
11. Barbara Dembin and Shuta Nakajima (2022), 'On the upper tail large deviation rate function for chemical distance in supercritical percolation', arXiv:2211.02605. https://arxiv.org/abs/2211.02605
12. Anne-Laure Basdevant, Nathanaël Enriquez, Lucas Gerin and Jean-Baptiste Gouéré (2014), 'The shape of large balls in highly supercritical percolation', Electronic Journal of Probability 19. DOI: 10.1214/ejp.v19-3062; arXiv:1305.0104. https://arxiv.org/abs/1305.0104
13. Chang-Long Yao (2021), 'Convergence of limit shapes for 2D near-critical first-passage percolation', arXiv:2104.01211. https://arxiv.org/abs/2104.01211
14. Chang-Long Yao (2023), 'Limit of the Wulff crystal when approaching criticality for isoperimetry in 2D percolation', Electronic Journal of Probability 28. DOI: 10.1214/23-ejp1061; arXiv:2303.04401. https://arxiv.org/abs/2303.04401
15. Marek Biskup, Oren Louidor, Eviatar B. Procaccia and Ron Rosenthal (2015), 'Isoperimetry in two-dimensional percolation', Communications on Pure and Applied Mathematics 68(9), 1483–1531. DOI: 10.1002/cpa.21558. https://doi.org/10.1002/cpa.21558
16. Christophe Garban, Gábor Pete and Oded Schramm (2018), 'The scaling limits of near-critical and dynamical percolation', Journal of the European Mathematical Society 20(4), 1195–1268. DOI: 10.4171/JEMS/786; arXiv:1305.5526. https://arxiv.org/abs/1305.5526
17. Michael Damron, Jack Hanson and Philippe Sosoe (2017), 'On the chemical distance in critical percolation', Electronic Journal of Probability 22. DOI: 10.1214/17-ejp88; arXiv:1506.03461. https://arxiv.org/abs/1506.03461
18. Irina Đanković, Maarten Markering, Jason Miller and Yizheng Yuan (2026), 'The scaling limit of random walk and the intrinsic metric on planar critical percolation', arXiv:2604.14122. https://arxiv.org/abs/2604.14122
19. Ioan Manolescu and Maran Mohanarangan (2026), 'The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality', arXiv:2603.16318. https://arxiv.org/abs/2603.16318
20. Stanislav Smirnov (2009), 'Critical percolation in the plane', arXiv:0909.4499 (a copy of the author's 2001 preprint establishing conformal invariance of critical site percolation on the triangular lattice). https://arxiv.org/abs/0909.4499