Fluid Wavefront & Geodesic Tree
Ready • Click lattice to add fluid source Flooded Cluster B(t)
Wavefront Frontier ∂B(t)
Shortest-Path Geodesic Ray
Fluid Source Node
Dry Lattice Edge
Lattice Parameters
Flooded Vertices
1
Simulation Time
0.00
Frontier Speed v(t)
0.00
Max Manhattan Radius
0
Flooding Speed v(t) vs Radius dr / dt
The Mathematical Frontier: Proved Theorems vs Open Conjectures
First-passage percolation (FPP), introduced by Broadbent & Hammersley (1965), is the standard model of fluid injection into random media. While fundamental global laws are now proved, microscopic boundary fluctuations remain one of modern probability's greatest challenges.
Rigorous Theorems (Proved)
The Asymptotic Shape Theorem (Cox–Durrett 1981)
There exists a deterministic convex compact set B0 ⊂ ℝd such that for any ε > 0, almost surely (1−ε)t B0 ⊆ B(t) ⊆ (1+ε)t B0 for all sufficiently large t. Randomness washes out on macroscopic scales.
Linear Growth Speed (Kingman's Subadditive Ergodic Theorem)
The passage time T(0, n·x) / n converges almost surely to a time constant μ(x). The asymptotic flooding speed along any ray is finite and strictly positive if and only if P(τ = 0) < pc.
The 2025 Five-Mathematician Speed Resolution
In late 2025, mathematicians resolved the sharp universal lower bound on flooding speed across disordered networks, establishing that bottleneck passages cannot decelerate the global wavefront below ballistic linear scaling.
Open Conjectures (Unproven)
KPZ Fluctuation Exponent in d = 2 (χ = 1/3, ξ = 2/3)
It is widely conjectured that Var(T(0, nx)) ~ n2/3, meaning wavefront fluctuations grow as n1/3. Proved for solvable models (corner growth), but unproved for standard FPP on ℤ2.
Strict Convexity of the Limit Ball B0
Does the limit shape have flat facets? It is conjectured that for continuous distributions like Exponential(1), the boundary ∂B0 is strictly convex with no flat edges, but a general proof remains elusive.
Fluctuations in High Dimensions (d ≥ 3)
Does the KPZ scaling relation 2χ = 2ξ − 1 persist in 3 dimensions and beyond? Fluctuation exponents in ℤ3 remain entirely open with conflicting numerical predictions.