Mean of the L^\∞-norm for L²-normalized random waves on compact\n aperiodic Riemannian manifolds

2013/10/04 by Yaiza Canzani, Boris Hanin, Canzani, Yaiza +1
Computer Science · Earth and Planetary Sciences · Mathematics · #35P20 (Primary) #58J37 #58J40 #58J50 #58J51 (Secondary) #Advanced Mathematical Modeling in Engineering #Arctic and Antarctic ice dynamics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1310.1361

openalex publication_date 2013/10/04 · openalex created_date 2022/08/28 · openalex updated_date 2026/07/28

Abstract

This article concerns upper bounds for L^\∞-norms of random approximate\neigenfunctions of the Laplace operator on a compact aperiodic Riemannian\nmanifold (M,g). We study f_\λ chosen uniformly at random from the\nspace of L²-normalized linear combinations of Laplace eigenfunctions with\neigenvalues in the interval (\λ², lr\λ+1²]. Our main result is\nthat the expected value of normf_\λ_\∞ grows at most like C\n\√(\log \λ) as \λ \→ \∞, where C is an explicit constant\ndepending only on the dimension and volume of (M,g). In addition, we obtain\nconcentration of the L^\∞-norm around its mean and median and study the\nanalogous problems for Gaussian random waves on (M,g).\n

Citations

• The spectral function of an elliptic operator

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