The uniform model is a reversible interactingparticle system that
evolves on the homogeneous tree. Occupied sites become vacant at rate one
provided the number of occupied neighbors does not exceed one.Vacant sites
become occupied at rate $\beta$ times the number of occupied neighbors. On the
binary tree, it has been shown that the survival threshold $\beta_c$ is 1/4. In
particular, for $\beta \leq 1/4$, the expected extinction time is
finite.Otherwise, the uniform model survives locally. We show that the survival
probability decays faster than a quadratic near $\beta_c$. This contrasts with
the behavior of the survival probability for the contact process on homogeneous
trees, which decays linearly.We also provide a lower bound that implies that
the rate of decay is slower than a cubic. Tools associated with reversibility,
for example, the Dirichlet principle and Thompson's principle, are used to
prove this result.