Open problem, Probability and statistics, posed 1957

Continuity of the phase transition in 3D percolation

Open

For Bernoulli bond (or site) percolation on $\mathbb{Z}^3$ with parameter $p$ , let $\theta(p) = \mathbb{P}_p(0 \leftrightarrow \infty)$ be the probability that the origin lies in an infinite open cluster, and let $p_c = \sup\{p : \theta(p) = 0\}$ be the critical probability. Is $\theta(p_c) = 0$ , i.e. is the percolation function continuous at $p_c$ ?

Research frontier as of 2026

As of 2026, continuity of $\theta$ at $p_c$ is proven for $d = 2$ (Kesten 1980 and related work) and for $d \ge 11$ via the lace expansion (Fitzner–van der Hofstad 2017, sharpening Hara–Slade's original $d \ge 19$ ). The physically relevant case $d = 3$ (and the range $3 \le d \le 10$ ) remains open; the March 2026 proof of supercritical sharpness on all infinite transitive graphs by Diskin, Easo, Radhakrishnan, Sudakov, and Tassion resolves a companion 1996 conjecture of Benjamini and Schramm but explicitly leaves the behavior exactly at $p_c$ on $\mathbb{Z}^3$ open, as emphasized in the paper's own discussion.

Best known results

Continuity of the phase transition ( $\theta(p_c) = 0$ ) is proven rigorously only for $d = 2$ (Kesten 1980, Harris 1960) and $d \ge 11$ (Fitzner–van der Hofstad 2017, via the lace expansion); the cases $3 \le d \le 10$ , including the physically important $d = 3$ , are open.
Sharpness of the phase transition (exponential decay for $p < p_c$ ) holds unconditionally in every dimension $d$ (Aizenman–Barsky and Menshikov 1987), and supercritical sharpness now holds on every infinite transitive graph (Diskin–Easo–Radhakrishnan–Sudakov–Tassion 2026), but neither result determines the value $\theta(p_c)$ itself.

Tools and where they stop

Tool	Achieved	Where it stops
Lace expansion (with non-backtracking refinement)	Rigorously proves mean-field critical exponents and continuity of $\theta$ at $p_c$ for nearest-neighbor percolation on $\mathbb{Z}^d$ , $d \ge 11$	The expansion coefficients fail to be summably small below $d \approx 10$ – $11$ , since low-dimensional lattices have too many short self-intersecting loops for the perturbative series to converge
Russo–Seymour–Welsh (RSW) theory and planar duality	Proves crossing probability estimates that pin down $p_c = 1/2$ and continuity of the phase transition for two-dimensional lattice percolation	Fundamentally relies on planar duality (interfaces between open and closed clusters are curves) and has no known analogue in three or more dimensions

Open questions

Is $\theta(p_c) = 0$ for Bernoulli bond percolation on $\mathbb{Z}^3$ ?
What is the smallest dimension $d$ below which the lace-expansion approach could in principle be pushed to prove continuity, and is $d = 6$ (the conjectured upper critical dimension) the true dividing line between mean-field and non-mean-field behavior?

References

Harry Kesten (1980). The critical probability of bond percolation on the square lattice equals 1/2 · DOI:10.1007/BF01221263
Michael Aizenman, David J. Barsky (1987). Sharpness of the phase transition in percolation models · DOI:10.1007/BF01212322
Robert Fitzner, Remco van der Hofstad (2017). Mean-field behavior for nearest-neighbor percolation in $d > 10$ · DOI:10.1214/17-EJP56 · arXiv:1506.07977
Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion (2026). Supercritical percolation on all infinite transitive graphs · arXiv:2603.03257 [preprint, not peer-reviewed]