= Log-stretch fluctuations in an isotropic renovating shear
{title2=$\operatorname{Var}(\log\lambda_N)=\frac N2\sum_{n\geq1}q^{2n}/n^2,\quad q^2=a^2/(a^2+4)$}
The logarithm of the stretch after $N$ independent uniform renovations is a sum of independent identically distributed bounded increments. For singular stretching $e^\gamma$, $\gamma=\operatorname{arsinh}(|a|/2)$, the increment has Fourier expansion $r+\sum_{n\geq1}(-1)^{n+1}q^n\cos(2n\theta)/n$, with $q=\tanh\gamma$ and $r=\tfrac12\log(1+a^2/4)$. Orthogonality proves the displayed variance. The <central limit theorem> gives an approximate central <log-normal distribution> of $\lambda_N$, not an exact description of extreme tails. For $|a|\ll1$, both the mean and variance of $\log\lambda_N$ are $Na^2/8$ to leading order. Direct expansion of fixed-power moments gives $\langle\lambda_N^p\rangle=\exp[Np(p+2)a^2/16+O(Na^4)]$; in particular mean stretch grows faster than typical stretch.
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