  Models of random walks are considered in which walkers are born at one
location and die at all other locations with uniform death rate. Steady-state
distributions of random walkers exhibit dimensionally dependent critical
behavior as a function of the birth rate. Exact analytical results for a
hyperspherical lattice yield a second-order phase transition with a nontrivial
critical exponent for all positive dimensions $D\neq 2,~4$. Numerical studies
of hypercubic and fractal lattices indicate that these exact results are
universal. Implications for the adsorption transition of polymers at curved
interfaces are discussed.
