Certified Locality in Diffusive Simulation: Tolerance Light Cones and Logarithmic Event Footprints

The heat equation propagates information at infinite speed, so a diffusive simulation cannot, in principle, ignore any cell. We show that once an output tolerance ε is fixed it can, with proof. For a point event of mass A in R^d, the set of points whose value ever exceeds ε is a ball of volume κ_d A/ε, with κ_d = (d/2e)^(d/2)/Γ(d/2 + 1); in the plane it is exactly A/(eε), independent of the diffusivity.

For an arbitrary event, a finite nonnegative measure of mass A, the footprint is at most C*_d A/ε with 1 ≤ C*_d ≤ min{3^d, C log(1 + d)}. Comparing footprints with the divisible sandpile, we prove that time-averaged footprints lie inside the sandpile set, of volume at most A/ε, in every dimension, and that 1 ≤ C*_1 ≤ 1.0433; an explicit example shows that the sandpile comparison cannot give less than 1.037. We conjecture C*_1 = C*_2 = 1.

With linear dissipation the footprint becomes polylogarithmic in A/ε. On the lattice, exponential supersolutions give windows valid for every reparametrization of the forcing, and a single theorem bounds the global error of a simulator that evolves only the union of its events' windows, retires events, and lets cells sleep between quantized announcements, by an explicit sum of budgets, uniformly in time and in the size of the domain; an event-driven scheduler attains exactly these values.

Computations confirm the continuum constants on the lattice to within 0.22%, and a certified 88-cell window reproduces a driven diffusion front with error 7.1 × 10^-3 against a certified bound of 1/2.

1. 1. Introduction1
2. 2. Setting4
3. 3. The tolerance light cone4
4. 4. Arbitrary events6
5. Certificates for arbitrary events6
6. The best constant7
7. 7. Dissipation and drift12
8. 8. Lattice certificates and certified simulation13
9. 9. Certificates in time16
10. 10. Numerical verification19
11. 11. Scope and open problems21