Directed Percolation with Colors and Flavors

A model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented. We use renormalized field theory and demonstrate that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the one-species Gribov process (Reggeon field theory). In addition this universal model shows an instability that generically leads to a total asymmetry between each pair of species of a cooperative society, and finally to unidirectionality of the interspecies couplings. It is shown that in general the universal multicritical properties of unidirectionally coupled directed percolation processes with linear coupling can also be described by the model. Consequently the crossover exponent describing the scaling of the linear coupling parameters is given by Φ=1 to all orders of the perturbation expansion. As an example of unidirectionally coupled directed percolation, we discuss the population dynamics of the tournaments of three species with colors of equal flavor.

[1]  M. Baker.  Higher order terms in the ϵ expansion of the pomeron propagator , 1974 .

[2]  Higher order ϵ-terms in the renormalization group approach to Reggeon field theory , 1974 .

[3]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[4]  P. Grassberger,et al.  Reggeon field theory and markov processes , 1978 .

[5]  L. Peliti,et al.  Field-theory renormalization and critical dynamics aboveTc: Helium, antiferromagnets, and liquid-gas systems , 1978 .

[6]  Victor Martin-Mayor,et al.  Field Theory, the Renormalization Group and Critical Phenomena , 1984 .

[7]  P. Schuster,et al.  On $\omega $-Limits for Competition Between Three Species , 1979 .

[8]  J. Coste,et al.  Asymptotic Behaviors in the Dynamics of Competing Species , 1979 .

[9]  S. Obukhov.  The problem of directed percolation , 1980 .

[10]  L. Peliti.  Path integral approach to birth-death processes on a lattice , 1985 .

[11]  T. Liggett.  Interacting Particle Systems , 1985 .

[12]  R. Ziff,et al.  Kinetic phase transitions in an irreversible surface-reaction model. , 1986, Physical review letters.

[13]  Ohtsuki,et al.  Nonequilibrium critical phenomena in one-component reaction-diffusion systems. , 1987, Physical review. A, General physics.

[14]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[15]  G. Grinstein,et al.  Critical phenomena in a nonequilibrium model of heterogeneous catalysis. , 1989, Physical review. A, General physics.

[16]  R. Dickman,et al.  Competing two-species directed percolation , 1991 .

[17]  Series analysis of the generalized contact process , 1994 .

[18]  Roughening transition in a one-dimensional growth process. , 1995, Physical review letters.

[19]  Nonequilibrium Critical Dynamics of a Three Species Monomer-Monomer Model. , 1996, Physical review letters.

[20]  `Real' versus `imaginary' noise in diffusion-limited reactions , 1997, cond-mat/9701069.

[21]  H. Janssen.  RENORMALIZED FIELD THEORY OF THE GRIBOV PROCESS WITH QUENCHED DISORDER , 1997 .

[22]  H. K. Janssen.  Spontaneous Symmetry Breaking in Directed Percolation with Many Colors: Differentiation of Species in the Gribov Process , 1997 .

[23]  Multicritical Behavior in Coupled Directed Percolation Processes. , 1997, cond-mat/9709057.

[24]  Equation of state for directed percolation , 1998, cond-mat/9810309.

[25]  J. Zinn-Justin.  Quantum Field Theory and Critical Phenomena , 2002 .