Random Lie bracket on $\mathfrak{sl}_2({\mathbf F}_p)$

Jezernik, Urban (avtor), Miščič, Matevž (avtor)

We study a random walk on the Lie algebra $\mathfrak{sl}_2({\mathbf F}_p)$ where new elements are produced by randomly applying adjoint operators of two generators. Focusing on the generic case where the generators are selected at random, we analyze the limiting distribution of the random walk and the speed at which it converges to this distribution. These questions reduce to the study of a random walk on a cyclic group. We show that, with high probability, the walk exhibits a pre-cutoff phenomenon after roughly $p$ steps. Notably, the limiting distribution need not be uniform, and it depends on the prime divisors of $p-1$. Furthermore, we prove that by incorporating a simple random twist into the walk, we can embed a well-known affine random walk on ${\mathbf F}_p$ into the modified random Lie bracket, allowing us to show that the entire Lie algebra is covered in roughly $\log p$ steps in the generic case.

• Jezik: Angleški jezik
• Ključne besede: random walks, Lie algebras, cyclic groups, random Lie bracket
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.01.2026
• Leto izida: 2026
• Št. strani: 19 str.
• Številčenje: Vol. 68, iss. 1, art. no. e70042
• PID: 20.500.12556/RUL-179129
• UDK: 512:519.2
• ISSN pri članku: 1042-9832
• DOI: 10.1002/rsa.70042
• COBISS.SI-ID: 267477251
• Datum objave v RUL: 05.02.2026

Gradivo je del revije

• Naslov: Random structures & algorithms
• Skrajšan naslov: Random struct. algorithms
• Založnik: J. Wiley
• ISSN: 1042-9832
• COBISS.SI-ID: 15158789

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.