Let \((X_ n)\) be a sequence of i.i.d. random matrices, with entries in a commutative, locally compact non-discrete field F, and let \(S_ n=X_ n....X_ 1\). When F is the real field, the theory of Lyapunov exponents of the random walk \(S_ n\) is well known. This theory has been extended to the ultrametric case and it then has applications to group theory.
A central point concerning these exponents is to determine whether they are distinct or not. This fact plays indeed a crucial role in the study of the integral representation of harmonic functions on the linear group. When F is the real field, one knows that under natural irreducibility assumptions the exponents are all distinct.
In this note, we extend this result to the case when F is a p-adic field, using techniques previously developed by Y. Guivarc’h [Ecole d’été de probabilites de Saint-Flour VIII-1978, Lect. Notes Math. 774, 117-250 (1980; Zbl 0433.60007); Probability measures on groups VII, Proc. Conf., Oberwolfach 1983, Lect. Notes Math. 1064, 161-181 (1984; Zbl 0561.60014)], and Y. Guivarc’h and A. Raugi [Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 187-242 (1985; Zbl 0558.60009)].