Dimension gap and phase transition for one-dimensional random walks with reflective boundary

We study $\mathbb Z$- and $\mathbb N$-extensions of interval maps with at most countably many full branches modelling one-dimensional random walks without and with a reflective boundary. We analyse the associated Gurevich pressure and explore the relations governing these two cases. For such extensions, we obtain variational formulae for the Gurevich pressure that depend only on the base system. As a consequence, we characterise the systems with a dimension gap and, in the presence of a reflective boundary, provide general conditions in terms of asymptotic covariances for a second order phase transition. As a by-product, we derive a variational formula for the spectral radius of infinite Hessenberg matrices.