7 research outputs found
Directed percolation with incubation times
We introduce a model for directed percolation with a long-range temporal
diffusion, while the spatial diffusion is kept short ranged. In an
interpretation of directed percolation as an epidemic process, this
non-Markovian modification can be understood as incubation times, which are
distributed accordingly to a Levy distribution. We argue that the best approach
to find the effective action for this problem is through a generalization of
the Cardy-Sugar method, adding the non-Markovian features into the geometrical
properties of the lattice. We formulate a field theory for this problem and
renormalize it up to one loop in a perturbative expansion. We solve the various
technical difficulties that the integrations possess by means of an asymptotic
analysis of the divergences. We show the absence of field renormalization at
one-loop order, and we argue that this would be the case to all orders in
perturbation theory. Consequently, in addition to the characteristic scaling
relations of directed percolation, we find a scaling relation valid for the
critical exponents of this theory. In this universality class, the critical
exponents vary continuously with the Levy parameter.Comment: 17 pages, 7 figures. v.2: minor correction
Epidemic processes with immunization
We study a model of directed percolation (DP) with immunization, i.e. with
different probabilities for the first infection and subsequent infections. The
immunization effect leads to an additional non-Markovian term in the
corresponding field theoretical action. We consider immunization as a small
perturbation around the DP fixed point in d<6, where the non-Markovian term is
relevant. The immunization causes the system to be driven away from the
neighbourhood of the DP critical point. In order to investigate the dynamical
critical behaviour of the model, we consider the limits of low and high first
infection rate, while the second infection rate remains constant at the DP
critical value. Scaling arguments are applied to obtain an expression for the
survival probability in both limits. The corresponding exponents are written in
terms of the critical exponents for ordinary DP and DP with a wall. We find
that the survival probability does not obey a power law behaviour, decaying
instead as a stretched exponential in the low first infection probability limit
and to a constant in the high first infection probability limit. The
theoretical predictions are confirmed by optimized numerical simulations in 1+1
dimensions.Comment: 12 pages, 11 figures. v.2: minor correction