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.
"Critical exponents for a reversible nearest particle system on the binary tree." Ann. Probab. 28 (1) 395 - 415, January 2000. https://doi.org/10.1214/aop/1019160124