Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929535835963392 |
|---|---|
| author | Michelen, Marcus Yakir, Oren |
| author_facet | Michelen, Marcus Yakir, Oren |
| contents | Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in $n$, and hence the energy is well-concentrated on the scale of $\sqrt{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02898 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere Michelen, Marcus Yakir, Oren Probability Mathematical Physics Classical Analysis and ODEs Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in $n$, and hence the energy is well-concentrated on the scale of $\sqrt{n}$. |
| title | Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere |
| topic | Probability Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2304.02898 |