Sign-balance of random Laplace eigenfunctions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910244400005120 |
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| author | Muirhead, Stephen Wigman, Igor |
| author_facet | Muirhead, Stephen Wigman, Igor |
| contents | Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Our results include the important case of random spherical harmonics, as well as more general band-limited random waves on smooth Riemannian manifolds. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above which random eigenfunctions are volume-balanced with respect to non-zero levels. Beyond their intrinsic interest, our results serve as a model for a natural conjecture on the optimal scale at which deterministic Laplace eigenfunctions are sign-balanced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22567 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sign-balance of random Laplace eigenfunctions Muirhead, Stephen Wigman, Igor Probability Mathematical Physics 60G60, 58J50 Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Our results include the important case of random spherical harmonics, as well as more general band-limited random waves on smooth Riemannian manifolds. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above which random eigenfunctions are volume-balanced with respect to non-zero levels. Beyond their intrinsic interest, our results serve as a model for a natural conjecture on the optimal scale at which deterministic Laplace eigenfunctions are sign-balanced. |
| title | Sign-balance of random Laplace eigenfunctions |
| topic | Probability Mathematical Physics 60G60, 58J50 |
| url | https://arxiv.org/abs/2604.22567 |