  The strong coupling between electronic transport in a single-level quantum
dot and a capacitively coupled nano-mechanical oscillator may lead to a
transition towards a mechanically-bistable and blocked-current state. Its
observation is at reach in carbon-nanotube state-of-art experiments. In a
recent publication [Phys. Rev. Lett. 115, 206802 (2015)] we have shown that
this transition is characterized by pronounced signatures on the oscillator
mechanical properties: the susceptibility, the displacement fluctuation
spectrum and the ring-down time. These properties are extracted from transport
measurements, however the relation between the mechanical quantities and the
electronic signal is not always straightforward. Moreover the dependence of the
same quantities on temperature, bias or gate voltage, and external dissipation
has not been studied. The purpose of this paper is to fill this gap and provide
a detailed description of the transition. Specifically we find: (i) The
relation between the current-noise and the displacement spectrum. (ii) The
peculiar behavior of the gate-voltage dependence of these spectra at the
transition. (iii) The robustness of the transition towards the effect of
external fluctuations and dissipation.
