Joint spectrum and large deviation principle for random matrix products

Abstract

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramér on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers S n = {g 1 . . . . .g n | g i ∈ S} of a subset S of a semisimple linear Lie group G (e.g. SL(d, R)). This limit set has applications, among others, in the study of large deviations.

Large deviation principle for random matrix products

“…Thus, D I is convex by Theorem 3.4. If G is semisimple, k = R and S is bounded, that int(D I ) = ∅ follows by 2. and the fact that in this case int(J(S)) = ∅ (see [12] or [26]). If S is unbounded, then we can find a bounded subset S 0 of S generating a Zariski dense sub-semigroup in G and such that µ(S 0 ) > 0.…”

Large deviations for random walks on free products of finitely generated groups

“…It is natural to expect the large deviation principle to hold for a large class of finitely generated groups, in particular for Gromov-hyperbolic groups; we expand slightly more on possible extensions of our approach 1 in this direction in Section 6. The applicability of the same strategy to such extensions, as well as to analogous questions in random matrix products, is already mentioned in [34].…”

“…Computing the exact expression of the rate function, in the cases treated by Theorem 1.4, is mostly out of reach; however, it is worth carrying through the computation in the easiest case of symmetric simple random walks on free groups, to get a flavour of what should happen in more general circumstances. This has already been performed in [33]: let G be a free group on r ≥ 1 generators, S = {a 1 , . .…”

The joint spectrum

“…The joint spectrum appears naturally in the context of random matrix products. In [80, 81], the second‐named author establishes a large deviation principle for independent and identically distributed (i.i.d.) random matrix products.…”

“…Then by Kingman's subadditive ergodic theorem, the following law of large numbers [12, 38, 43] holds almost surely: The right‐hand side is called the Lyapunov vector of the ergodic process and clearly lies in . In [80, 81], the second‐named author proves a large deviation principle for the Cartan projection of i.i.d. random walks supported on .…”