  We propose and investigate numerically a one-dimensional model which exhibits
a non-Anderson disorder-driven transition. Such transitions have recently been
attracting a great deal of attention in the context of Weyl semimetals,
one-dimensional systems with long-range hopping and high-dimensional
semiconductors. Our model hosts quasiparticles with the dispersion $\pm
|k|^\alpha \mathrm{sign} k$ with $\alpha<1/2$ near two points (nodes) in
momentum space and includes short-range-correlated random potential which
allows for scattering between the nodes and near each node. In contrast with
the previously studied models in dimensions $d<3$, the model considered here
exhibits a critical scaling of the Thouless conductance which allows for {an
accurate} determination of the critical properties of the non-Anderson
transition, with a precision significantly exceeding the results obtained from
the critical scaling of the density of states, usually simulated at such
transitions. We find that in the limit of the vanishing parameter
$\varepsilon=2\alpha-1$ the correlation-length exponent
$\nu=2/(3|\varepsilon|)$ at the transition is inconsistent with the prediction
$\nu_{RG}=1/|\varepsilon|$ of the perturbative renormalisation-group analysis.
Our results allow for a numerical verification of the convergence of
$\varepsilon$-expansions for non-Anderson disorder-driven transitions and, in
general, interacting field theories near critical dimensions.
