  We consider the Glauber dynamics for model of polymer interacting with a
substrate or wall. The state space is the set of one-dimensional
nearest-neighbor paths on $\mathbb{Z}$ with nonnegative integer coordinates,
starting at $0$ and coming back to $0$ after $L$ ($L\in 2\mathbb{N}$) steps and
the Gibbs weight of a path $\xi=(\xi_x)^{L}_{x=0}$ is given by
$\lambda^{\mathcal{N}(\xi)}$, where $\lambda \geq 0$ is a parameter which
models the intensity of the interaction with the substrate and
$\mathcal{N}(\xi)$ is the number of zeros in $\xi$. The dynamics we consider
proceeds by updating $\xi_x$ with rate one for each $x=1,\dots, L-1$, in a
heat-bath fashion. This model was introduced in [CMT08] with the aim of
studying the relaxation to equilibrium of the system.
  We present new results concerning the total variation mixing time for this
dynamics when $\lambda< 2$, which corresponds to the phase where the effects of
the wall's entropic repulsion dominates. For $\lambda \in [0, 1]$, we prove
that the total variation distance to equilibrium drops abruptly from $1$ to $0$
at time $(L^2 \log L)(1+o(1))/\pi^2$. For $\lambda \in (1,2)$, we prove that
the system also exhibit cutoff at time $(L^2 \log L)(1+o(1))/\pi^2$ when
considering mixing time from "extremal conditions" (that is, either the highest
or lowest initial configuration for the natural order on the set of paths). Our
results improves both previously proved upper and lower bounds in [CMT08].
