The Continuum Limit of Some Products of Random Matrices Associated with Renewing Flows

Abstract We consider the continuum limit of some products of random matrices in $$\text {SL}(d,{\mathbb {R}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>SL</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that arise as discretisations of incompressible renewing flows— that is, of flows corresponding to a divergence-free velocity field that takes independent, identically-distributed values in successive time intervals of duration proportional to $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> . The statistical properties of the product are encoded in its generalised Lyapunov exponent whose computation reduces to finding the leading eigenvalue of a certain transfer operator. In the continuum limit obtained by neglecting the terms of order $$o(\tau ^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the transfer operator becomes a partial differential operator and, for a certain type of disorder which we call “symmetric”, some calculations are feasible. For $$d=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we compute the growth rate of the product in terms of complete elliptic integrals. By letting the elliptic modulus vary, we obtain a spectral problem, corresponding to a modulus-dependent random renewing flow, which may be viewed as a perturbation of the spectral problem for the angular Laplacian. In this way, we deduce expansions for the generalised Lyapunov exponent in ascending powers of the modulus. These expansions generalise to the case $$d \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and we compute the first few terms explicitly for $$d \in \{2,3\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> .