Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.01866 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914363279933440 |
|---|---|
| author | Barbieri, Marco Lekše, Marusa Zozaya, Andoni |
| author_facet | Barbieri, Marco Lekše, Marusa Zozaya, Andoni |
| contents | We obtain explicit upper and lower bounds for the expected action energy associated with a pair $({\sf A},{\sf Δ})$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $({\sf A},{\sf A})$ and $({\sf A},{\sf A}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $|{\sf A}^{\ast 2}|$ and $|{\sf A}{\sf A}^{-1}|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01866 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the expected value of energy in groups Barbieri, Marco Lekše, Marusa Zozaya, Andoni Group Theory Combinatorics Probability primary: 20P05, secondary: 05E16, 11B13 We obtain explicit upper and lower bounds for the expected action energy associated with a pair $({\sf A},{\sf Δ})$ of subsets sampled uniformly at random from a permutation group and its domain, respectively. We then specialize these bounds to multiplicative energy in several settings. In particular, we derive sharp asymptotic formulae for the expected energy of pairs of the form $({\sf A},{\sf A})$ and $({\sf A},{\sf A}^{-1})$. Finally, we apply these estimates to derive probabilistic results on the existence of subsets with large growth and to compare the typical behaviour of the cardinalities of the sets $|{\sf A}^{\ast 2}|$ and $|{\sf A}{\sf A}^{-1}|$. |
| title | On the expected value of energy in groups |
| topic | Group Theory Combinatorics Probability primary: 20P05, secondary: 05E16, 11B13 |
| url | https://arxiv.org/abs/2603.01866 |