For every delta>0 there exists varepsilon>0 such that the following holds uniformly over every finite graph G, every percolation parameter p∈[0,1], every origin vertex o∈ V(G), every boundary set partial G⊆ V(G), and every collection of increasing events (A_v)_v∈partial G satisfying P_p(A_v)≥ 1-varepsilon for all v∈partial G: P_p!(exists v∈partial G: o longleftrightarrow v and A_v middle| o longleftrightarrowpartial G)≥ 1-delta. Here o longleftrightarrowpartial G denotes the event that o is connected by open edges to some vertex of partial G. The conjecture would imply theta(p_c)=0 for Bernoulli bond percolation on Z^d for every d≥ 2.

The conjecture implying θ(p_c)=0 for Bernoulli bond percolation on Z^d

For every $\delta>0$ there exists $\varepsilon>0$ such that the following holds uniformly over every finite graph $G$ , every percolation parameter $p\in[0,1]$ , every origin vertex $o\in V(G)$ , every boundary set $\partial G\subseteq V(G)$ , and every collection of increasing events $(A_v)_{v\in\partial G}$ satisfying $\mathbb{P}_p(A_v)\geq 1-\varepsilon$ for all $v\in\partial G$ : $\mathbb{P}_p\!\left(\exists v\in\partial G:\ o\longleftrightarrow v\ \text{and}\ A_v\ \middle|\ o\longleftrightarrow\partial G\right)\geq 1-\delta$ . Here $o\longleftrightarrow\partial G$ denotes the event that $o$ is connected by open edges to some vertex of $\partial G$ . The conjecture would imply $\theta(p_c)=0$ for Bernoulli bond percolation on $\mathbb{Z}^d$ for every $d\geq 2$ .

References

Primary source

arXiv

Additional references

A conjecture implying θ(p_c)=0 for Bernoulli bond percolation on Z^d — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper reduces the all-dimensional critical-percolation question to a finite-graph conjecture and proves a substantial planar case, but does not settle the original question.

Benjamini and Schramm posed the broader conjecture in 1996 that critical percolation has no infinite cluster on every transitive graph. The present problem asks for the finite-graph inequality whose weaker form would imply $\theta(p_c)=0$ on $\mathbb{Z}^d$ for every $d\geq 2$ .

Known results

Planar bond percolation: Kesten, 1980.
Planar site percolation: Russo, 1981.
High-dimensional mean-field result: Hara and Slade, 1990.
Thickness-two and finite-slab extensions: Damron, Newman and Sidoravicius, 2015; Duminil-Copin, Sidoravicius and Tassion, 2016.

September 2026 finite-graph development

The new paper reports the finite-graph conjecture, proves it for finite planar graphs with outer-face boundary vertices, obtains the explicit bound $1-2\sqrt{\varepsilon}$ , and gives a counterexample for arbitrary increasing events without connectivity structure. It states that the central conjecture remains neither proved nor disproved; consequently, the reduction is conditional rather than a proof of $\theta(p_c)=0$ .

Current status (as of September 2026): The planar finite-graph case and the conditional reduction are reported, but the central finite-graph conjecture and $\theta(p_c)=0$ for general $\mathbb{Z}^d$ , notably $d=3$ , remain open.

Sources

No solutions have been posted yet.