Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

Alexander I. Bufetov (1, 2, 3, 4) , Yanqi Qiu (5)

1 I2M - Institut de Mathématiques de Marseille
2 SMI | RAS - Steklov Mathematical Institute [Moscow]
3 Institute for Information Transmission Problems
4 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
5 IMT - Institut de Mathématiques de Toulouse UMR5219

Let $F$ be a non-discrete non-Archimedean locally compact field and $\mathcal{O}_F$ the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Mat}(\mathbb{N}, F)$ of infinite matrices with enties in $F$ with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F) \times \mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.6 that, for non-dyadic $F$, classifies ergodic probability measures on the space $\mathrm{Sym}(\mathbb{N}, F)$ of infinite symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F)$.

Mots clés: Ismagilov-Ols hanski multi- plicativity; ergodic measures; random matrices; non-Archimedean locally compact fields; Vershik-Kerov ergodic method; orbital integrals; inductively compact groups

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Systèmes dynamiques [math.DS]; Combinatoire [math.CO]; Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

Alexander I. Bufetov, Yanqi Qiu. Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields. Compositio Mathematica, 2017, 153 (12), pp.2482-2533. ⟨10.1112/S0010437X17007412⟩. ⟨hal-01483629⟩