  We consider the quantum first detection problem for a particle evolving on a
graph under repeated projective measurements with fixed rate $1/\tau$. A
general formula for the mean first detected transition time is obtained for a
quantum walk in a finite-dimensional Hilbert space where the initial state
$|\psi_{\rm in}\rangle$ of the walker is orthogonal to the detected state
$|\psi_{\rm d}\rangle$. We focus on diverging mean transition times, where the
total detection probability exhibits a discontinuous drop of its value, by
mapping the problem onto a theory of fields of classical charges located on the
unit disk. Close to the critical parameter of the model, which exhibits a
blow-up of the mean transition time, we get simple expressions for the mean
transition time. Using previous results on the fluctuations of the return time,
corresponding to $|\psi_{\rm in}\rangle = |\psi_{\rm d}\rangle$, we find close
to these critical parameters that the mean transition time is proportional to
the fluctuations of the return time, an expression reminiscent of the Einstein
relation.
