Gaussian holomorphic sections on noncompact complex manifolds

2023/02/16 by Drewitz, Alexander, Liu, Bingxiao, Marinescu, George · 2 citations
#32A60 #32U40 #53D50 #60B12 #60D05 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2302.08426

Abstract

We give two constructions of Gaussian-like random holomorphic sections of a Hermitian holomorphic line bundle (L,h_L) on a Hermitian complex manifold (X,Θ). In particular, we are interested in the case where the space of L²-holomorphic sections H⁰₍₂₎(X,L) is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if dim H⁰₍₂₎(X,L)=∞, are almost never L²-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin-Toeplitz quantization and yields a random L²-holomorphic section. Furthermore, we study their random zeros in the context of semiclassical limits, including their equidistribution, large deviation estimates and hole probabilities.

Cited by

• Asymptotics of hole probability regarding open balls for random sections on compact Riemann surfaces
• On the supremum of random cusp forms

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