Nonperturbative renormalization group for the diffusive epidemic process

We consider the Diffusive Epidemic Process (DEP), a two-species reaction-diffusion process originally proposed to model disease spread within a population. This model exhibits a phase transition from an active epidemic to an absorbing state without sick individuals. Field-theoretic analyses suggest that this transition belongs to the universality class of Directed Percolation with a Conserved quantity (DP-C, not to be confused with conserved-directed percolation C-DP, appearing in the study of stochastic sandpiles). However, some exact predictions derived from the symmetries of DP-C seem to be in contradiction with lattice simulations. Here we revisit the field theory of both DP-C and DEP. We discuss in detail the symmetries present in the various formulations of both models. We then investigate the DP-C model using the derivative expansion of the nonperturbative renormalization group formalism. We recover previous results for DP-C near its upper critical dimension $d_c=4$, but show how the corresponding fixed point seems to no longer exist below $d\lesssim 3$. Consequences for the DEP universality class are considered.

Figure 1

Values of $\eta$ as a function of the dimension $d$ for $\mu =0$ calculated from the ${\mathrm{LPA}}^{\prime }$ truncations compared to the one-loop result (dashed line) and to the simulation [12] (isolated point at $d=1$). Thick saltires mark the disappearance of the fixed point.

Figure 2

Values of $\overline{\eta }$ as a function of the dimension for (a) $\mu =-1$ and (b) $\mu =0.5$ calculated from the ${\mathrm{LPA}}^{\prime }$ truncations compared to the one-loop result (dashed line). Thick saltires mark the disappearance of the fixed point.