0.434 \cdot \log \sqrt{\sigma^{-2}}$. The dashed line represents the numerical zero $\log(2N)/N$.[]{data-label="fig:gammaOfsigma"}](GammaOfSigma){width="\linewidth"}

The equilibrium state is essentially an eigenvector of $L$ to its zero eigenvalue; therefore it is interesting to look at the eigenvalue localization properties of $L$ also for other eigenvalues $\lambda$. The corresponding measure of localization is given by the Lyapunov exponent $\gamma$ which is defined as follows: Let $\pi_k$ be the components of an eigenvector of $L$ to eigenvalue $\lambda$. Then the following relation holds: $$\begin{aligned}
    \begin{pmatrix}
       \frac{\pi_k}{\tau_k}{
  \parbox[0pt][2em][c]{0cm}{}
}\\ \frac{\pi_{k+1}}{\tau_{k+1}}{
  \parbox[0pt][2em][c]{0cm}{}
}
    \end{pmatrix}
    &= \underbrace{\begin{pmatrix}
      0 & 1\\ -1 & 2 - \lambda \tau_k
       \end{pmatrix}}_{=~T_k}
    \begin{pmatrix}
    \frac{\pi_{k-1}}{\tau_{k-1}}{
  \parbox[0pt][2em][c]{0cm}{}
}\\ \frac{\pi_k}{\tau_k}{
  \parbox[0pt][2em][c]{0cm}{}
}
    \end{pmatrix}.
    \label{equ:Tk2}\end{aligned}$$ The matrix $T_k$ which transfers between the vector on the r.h.s. and the vector on the l.h.s. is called the $k$th transfer matrix. The Lyapunov exponent is now defined as $$\begin{aligned}
     \gamma = \lim_{N \rightarrow \infty} \frac{1}{N} \left\langle \log \| T_N \ldots T_2 T_1 \|_1\right\rangle_\tau,
     \label{equ:Defgamma}\end{aligned}$$ with $\| M \|_1 = \max_{j = 1,2} \left( |m_{1j}| + |m_{2j}| \right)$ denoting the maximum absolute column sum of the matrix $M$. Eq. (\[equ:Defgamma\]) together with Eq. (\[equ:Tk2\]) defines $\gamma(\lambda)$ regardless of the fact whether $\lambda$ is an eigenvalue of $L$ or not. (Note that 