Math Conjectures

KPZ fluctuation exponent for two-dimensional first-passage percolation

OPENLandmarkConjectureProposed c. 1986 · Standard version

Canonical statement

Give each unoriented nearest-neighbor edge ee of Z2\mathbb Z^2 an independent exponential random variable τe\tau_e of mean 11. For x,y∈Z2x,y\in\mathbb Z^2, define
T(x,y)=min⁡π:x→y∑e∈πτe, T(x,y)=\min_{\pi:x\to y}\sum_{e\in\pi}\tau_e,
where the minimum is over finite nearest-neighbor paths from xx to yy. If e1=(1,0)e_1=(1,0), then
lim⁡n→∞log⁡Var⁡T(0,ne1)log⁡n=23. \lim_{n\to\infty}\frac{\log\operatorname{Var}T(0,ne_1)}{\log n}=\frac23.
View source LaTeX
Give each unoriented nearest-neighbor edge \(e\) of \(\mathbb Z^2\) an independent exponential random variable \(\tau_e\) of mean \(1\). For \(x,y\in\mathbb Z^2\), define
\[
T(x,y)=\min_{\pi:x\to y}\sum_{e\in\pi}\tau_e,
\]
where the minimum is over finite nearest-neighbor paths from \(x\) to \(y\). If \(e_1=(1,0)\), then
\[
\lim_{n\to\infty}\frac{\log\operatorname{Var}T(0,ne_1)}{\log n}=\frac23.
\]
PROB-002:[HammersleyWelsh1965FPP][AuffingerDamronHanson2017FPP][DamronHansonSosoe2015]
Probability theoryDynamical systemsMathematical physics

Notes

First-passage percolation, introduced by Hammersley and Welsh in 1965, attaches independent random weights to the edges of Z2\mathbb Z^2 and studies the induced random metric TT, the least total weight of a path between two vertices [HammersleyWelsh1965FPP]. The subadditive ergodic theorem shows that T(0,ne1)T(0,ne_1) grows linearly in nn. The conjecture, reflecting the Kardar–Parisi–Zhang scaling picture that emerged in the mid-1980s physics literature, concerns the fluctuations: with mean-one exponential edge weights, Var⁡T(0,ne1)\operatorname{Var}T(0,ne_1) should grow like n2/3n^{2/3} on a logarithmic scale, corresponding to passage-time fluctuations of order n1/3n^{1/3}.

Rigorous knowledge is much weaker. The variance is known to be sublinear in nn for a broad class of edge-weight distributions [DamronHansonSosoe2015], and the 2/32/3 exponent has been proved in exactly solvable directed last-passage models believed to share the same universality class; the state of the field is surveyed by Auffinger, Damron and Hanson [AuffingerDamronHanson2017FPP].

For genuine undirected first-passage percolation the exponent remains unproved even in this fixed exponential model, and the problem is open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.