Where the Euler characteristic ends and Pickands begins

A centered, unit-variance stationary Gaussian field on the unit square with covariance r(t) = exp(−(|t₁|^α + |t₂|^α)/ℓ^α), so that r(t) ≈ 1 − (|t₁|^α+|t₂|^α)/ℓ^α near the origin. For α = 2 the field is smooth and the excursion set above a high level is a few clean islands: the Euler-characteristic heuristic of Adler–Taylor (the one in Pavel Ievlev's illustration) gives P(sup X > u) ≈ E χ(A_u). For α < 2 the field is not mean-square differentiable, so this smooth-field Euler-characteristic formula does not apply. Excursion boundaries become rougher, while the high-level tail obeys the Pickands–Piterbarg asymptotics P(sup X > u) ~ ℋ_α² ℓ⁻² u^4/α Ψ(u) as u → ∞, where Ψ(u) = P(N(0,1) > u). The Pickands constant ℋ_α has the classical explicit values for α = 1 (ℋ₁ = 1) and α = 2 (ℋ₂ = 1/√π).

Gaussian field sampled on a 100×100 grid

grid excursion {X > u}: — components (side-sharing pixels)

Monte Carlo P(grid max X > u)

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press “Run Monte Carlo”

Smooth-field E χ(A_u)

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Finite-level implied ℋ_α

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Roughness α = 2.00 Level u = 1.75 Correlation length ℓ = 0.25

Monte Carlo: 0 independent Gaussian fields on a 56×56 grid, sampled from the stated covariance by numerical factorisation. The level slider re-reads their maxima instantly. A grid maximum can miss continuous peaks, especially at small α; Monte Carlo sampling and finite levels add further error. The implied constant is simply √( P_MC / (ℓ⁻² u^4/α Ψ(u)) ): what the simulation would need ℋ_α to be. This is a finite-level illustration, not a consistent estimator of ℋ_α. The smooth-field expected Euler characteristic can exceed 1 at moderate levels; it is a probability approximation only at high levels.

The field near a conditioned high value

For a one-dimensional slice, condition on X(0) = u and plot u[X(ℓ u^−2/αs) − u]. As u → ∞, this converges on compact intervals to √2 B_α(s) − |s|^α, where B_α has Hurst index α/2 and variance |s|^α. At α = 2 the limit is a parabola with a Gaussian linear tilt. The plot samples the finite-level conditional Gaussian law at the selected u; the dashed curve is the limiting mean. The origin is a conditioned point, so nearby values can exceed u.

Mathematical background: Taylor–Takemura–Adler: the smooth-field EC approximation; Dębicki–Hashorva–Soja-Kukieła: homogeneous Gaussian fields; Michna: Pickands’ theorem and conditional scaling.