The dying percolation conjecture (also known as θ ( p c ) = 0 {\displaystyle \theta (p_{c})=0} conjecture) is a prominent conjecture in percolation theory. It states that critical Bernoulli bond percolation on Z d {\displaystyle \mathbb {Z} ^{d}} for d ≥ 2 {\displaystyle d\geq 2} almost surely has no infinite clusters. The conjecture is proved for the cases d = 2 {\displaystyle d=2} by Kesten, for d ≥ 19 {\displaystyle d\geq 19} by Hara and Slade using the technique called "lace expansion". The later result was then improved to d ≥ 11 {\displaystyle d\geq 11} by Fitzner and van der Hofstad. The conjecture is sometimes called the "Dying percolation conjecture", a name used by mathematician Gil Kalai. The name evokes the intuition that the cluster at origin "just barely" fails to survive at the critical threshold.
Background In nearest-neighbor Bernoulli bond percolation on Z d {\displaystyle \mathbb {Z} ^{d}} , each edge of the integer lattice is independently declared open with probability p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} and closed otherwise. A cluster is a maximal connected subgraph of open edges. The central question of percolation theory is: for which values of p {\displaystyle p} does an infinite cluster exist almost surely. The percolation probability θ ( p ) {\displaystyle \theta (p)} is defined as the probability that the cluster containing 0 {\displaystyle 0} is infinite. By a coupling argument, one can show that θ {\displaystyle \theta } is a non-decreasing function of p {\displaystyle p} . As a corollary, for any dimension d {\displaystyle d} there exists a critical probability (denoted by p c {\displaystyle p_{c}} ) , such that
For p < p c {\displaystyle p<p_{c}} , every cluster is almost surely finite. For p > p c {\displaystyle p>p_{c}} , an infinite cluster almost surely exists. The conjecture asks whether θ ( p c ) = 0 {\displaystyle \theta (p_{c})=0} , i.e. whether θ {\displaystyle \theta } is continuous or has a discontinuity at p c {\displaystyle p_{c}} .
Equivalent formulations The following statements are equivalent for Bernoulli percolation on Z d {\displaystyle \mathbb {Z} ^{d}} :
Bernoulli bond percolation with critical parameter p c {\displaystyle p_{c}} has no infinite cluster almost surely; No infinite cluster at criticality at the origin: θ ( p c ) = 0 {\displaystyle \theta (p_{c})=0} ; Continuity of θ {\displaystyle \theta } at p c {\displaystyle p_{c}} ; Continuity of θ {\displaystyle \theta } . The last formulation is equivalent to the others, since Aizenman, Kesten and Newman showed that θ {\displaystyle \theta } is continuous at all points of [ 0 , 1 ] {\displaystyle [0,1]} except for possibly p c {\displaystyle p_{c}} .
Significance The θ ( p c ) = 0 {\displaystyle \theta (p_{c})=0} conjecture is among the central open problems in percolation theory. Benjamini et al. (1999) open their paper by describing it as "the main long-standing open question in percolation theory". Fields medalist Hugo Duminil-Copin in his ICM lecture "Sixty years of percolation" frames the development of percolation theory around this conjecture. Louigi Addario-Berry suggested that it's among the most important problems in the larger field of probability theory. This conjecture needs to be true for most percolation critical exponents to be well-defined. As such, it is predicted by physicists studying phase transition theory. The most relevant physically case ( d = 3 {\displaystyle d=3} ) remains open.
Known results
d=1 For d = 1 {\displaystyle d=1} , the critical probability p c = 1 {\displaystyle p_{c}=1} and θ ( p ) = { 0 , if p < 1 1 , if p = 1. {\displaystyle \theta (p)={\begin{cases}0,{\text{ if }}p<1\\1,{\text{ if }}p=1.\end{cases}}}
So the conjecture is false. However, for d ≥ 2 {\displaystyle d\geq 2} , p c ∈ ( 0 , 1 ) {\displaystyle p_{c}\in (0,1)} .., and the conjecture is widely believed to be true.
d=2 The dimension d = 2 {\displaystyle d=2} is the only dimension (besides 1) where p c {\displaystyle p_{c}} is known exactly. For the other dimensions, there are only numerical approximations. It was proven to be 1 2 {\displaystyle {\frac {1}{2}}} by Kesten after years of being the problem that drove most of the efforts in the field. This resolved the conjecture for d = 2 {\displaystyle d=2} , since θ ( 1 2 ) = 0 {\displaystyle \theta \left({\frac {1}{2}}\right)=0} was already proven by Harris. For 2-dimensional lattices, continuity of the phase transition is proved only for some lattices other than square lattice. It is conjectured for them to belong to the same universality class and so to have the same critical behavior.
High dimensions ( d ≥ 11 {\displaystyle d\geq 11} ) For larger d {\displaystyle d} , the conjecture is listed as Problem 5 in Section 12 of Kesten (1982). Since then, the partial cases and generalizations of the conjecture received much attention In 1990, Hara and Slade used the lace expansion technique to show that percolation at the critical parameter p c {\displaystyle p_{c}} exhibits a mean-field behavior. Namely, they showed that quantities such as θ {\displaystyle \theta } obey the power laws, with exponents predicted by the mean-field theory. Later, their techniques were improved to lower the dimension down to 11. However, the mean-field behavior cannot hold in dimensions 5 and less, so dimensions 3, 4 and 5 cannot be covered by an extension of this proof. For dimensions higher than 6, the conjecture is true for a "spread out" percolation, where additional edges are added to any pair of vertices on distance less than a large enough parameter L {\displaystyle L}
Percolation on half-space It is shown in Barsky, Grimmett & Newman (1991) that for any d ≥ 2 {\displaystyle d\geq 2} , the critical bond percolation restricted to a half-space N × Z d − 1 {\displaystyle \mathbb {N} \times \mathbb {Z} ^{d-1}} almost surely doesn't contain an infinite cluster. This is in contrast with the result of Grimmett & Marstrand (1990) that for any p > p c {\displaystyle p>p_{c}} , the percolation on a slab [ 0 , n ] d − 2 × Z 2 {\displaystyle [0,n]^{d-2}\times \mathbb {Z} ^{2}} almost surely has an infinite cluster for large enough n {\displaystyle n} .
Nonamenable and exponential-growth graphs The conjecture has been verified for a large class of transitive graphs whose growth is faster than any polynomial. The foundational result in this direction is due to Benjamini et al. (1999): critical Bernoulli percolation on any Cayley graph of a nonamenable group has no infinite clusters. As a trivial partial case, it covers infinite d {\displaystyle d} -regular trees. In Hutchcroft (2016), this result was extended to all quasi-transitive graphs of exponential growth.
Reduction to a conjectural correlation inequality In Kozma & Nitzan (2024), the authors reduced the conjecture to various correlation inequalities decreasing in strength. They are supported by numerical evidence, and if true, any of them will resolve the conjecture.
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