Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices

Abstract

Abstract. We consider a certain infinite product of random 2×2 matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter ε > 0 and on a real random variable with distribution µ. For a large class of µ, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16, 1641Phys. A 16, -2654Phys. A 16, (1983) that the Lyapunov exponent behaves like Cε 2α in the limit ε 0, where α ∈ (0, 1) and C > 0 are determined by µ. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small ε. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.

Regular Expansion for the Characteristic Exponent of a Product of 2 × 2 Random Matrices

“…Then, they use this probability to compute the Lyapunov exponent. This two scale analysis is made rigourous by G. Genovese et al [12], who show that this probability measure is indeed close to the invariant measure in a suitable norm, and this control is sufficiently strong to yield precisely (1.7). It appears to be rather challenging to follow the same steps for α 1: the guess for the invariant probability would have to be tuned to yield the ⌊α⌋ terms of the regular expansion and the singular ǫ 2α term.…”

“…The neat thing about that theorem is that, unlike the lower bound (1.16) of Theorem A, the estimate (1.21) should be "sharp" in the following sense. If one proves that, as ǫ goes to 0, P(X ǫ cǫ −2 ) Cǫ 2α for some positive constants c and C (the precise analysis of M ǫ 's invariant measure conducted in [12] provides such an estimate when α ∈ (0, 1)) then (1.21) becomes cǫ 2α R K (ǫ) Cǫ 2α (with a log correction if α is an integer). It is the good order of ǫ predicted by Conjecture 1.2.…”

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

“…The assumption on the values of Z is important: in [12] an example without this property is given in which C Z must be replaced by a log-periodic function is given. In [14] (1.3) has been proven under the stronger assumption that the law of Z has a compact support bounded away from zero and that Z has a C 1 density. Beyond the application, this result identifies a singular behavior of the Lyapunov exponents and enters the field of inquiry into the regularity of Lyapunov exponents (e.g.…”

“…We anticipate that our results will hold asymptotically under the assumption that Z has a suitably regular density and without requiring the support to be bounded (but both Z and 1/Z are in L p for a p > 1). As in [14], we will start from the 2-scale idea of [12]. What is done in [12] is to guess a probability measure -we call it the DH probability -that should be sufficiently close to the Furstenberg probability (on the projective space, i.e.…”

Continuum Limit of Random Matrix Products in Statistical Mechanics of Disordered Systems

“…The arguments in [12] aim at constructing a probability that for ε small is expected to be close to the invariant probability. In [20] this construction is put on rigorous grounds and, above all, it is shown that this probability, although not invariant, is sufficiently close to the invariant one to make it possible to control the Lyapunov exponent with the desired precision. A result about (1.26), i.e.…”