2024/12/10 by Fu, Qiangang "Brandon'', Nicolaescu, Liviu I.
#60D05 #60F15 #60F17 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.07690
On the flat torus \mathbbTm=ℝm/ℤm with angular coordinates θ we consider the random function FR=\mathfraka( R-1 √Δ ) W, where R>0, Δ is the Laplacian on this flat torus, \mathfraka is an even Schwartz function on ℝ such that \mathfraka(0)>0 and W is the Gaussian white noise on \mathbbTm viewed as a random generalized function. For any f∈ C(\mathbbTm) we set ZR(f):=∑∇ FR(θ)=0 f(θ) We prove that if the support of f is contained in a geodesic ball of \mathbbTm, then the variance of ZR(f) is asymptotic to const× Rm as R→∞. We use this to prove that if m≥ 2, then as N→∞ the random measures N-mZN(-) converge a.s. to an explicit multiple of the volume measure on the flat torus.