The Small-Radius Euler Characteristic of a Smooth Gaussian Tube via Kac–Rice

We study the small-radius topology of the Euclidean tube generated by a smooth planar Gaussian path. Let X=(Xt)0≤t≤T be a sufficiently regular Gaussian process with values in R2, and define its radius-ε tube by Tε={x∈R2:dist(x,X([0,T]))≤ε}. When the sample path is a smooth immersed curve with finitely many transversal double points and no higher-order intersections, the tube is, for all sufficiently small radii, a regular neighbourhood of the finite planar graph traced by the curve. Consequently, its Euler characteristic is determined by the number N of self-intersections: χ(Tε)=1−N for all sufficiently small ε, almost surely. We combine this deterministic topological observation with the Kac–Rice formula applied to the two-parameter Gaussian difference field F(s,t)=Xt−Xs. This yields an exact integral expression for the expected limiting Euler characteristic E[χ0], where χ0=limε↓0χ(Tε). For stationary Gaussian coordinates with covariance r, the formula reduces to a one-dimensional integral depending only on r, r′, and r″. We then specialize to the squared-exponential covariance r(u)=exp(−u2/2ℓ2), obtaining a fully explicit dimensionless quadrature depending only on T/ℓ. For the squared-exponential kernel, the expected number of limiting small-radius holes satisfies E[N]∼18π(T/ℓ)2 in the short-correlation regime. Finally, we add a constant linear drift and show that the expected number of self-intersections is modified by two competing mechanisms: a Gaussian density-damping term and a noncentral velocity-amplification term.