No abstract

Healthy and sick individuals (A and B particles) diffuse independently with diffusion constants D A and D B . Sick individuals upon encounter infect healthy ones (at rate k), but may also spontaneously recover (at rate 1/τ ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρ c of the spatially averaged total density. The epidemic evolves as the diffusion-reaction-decay process A + B → 2B, B → A, for which we write down the field theory. The stationary state properties of this theory when D A = D B were obtained by Kree et al. The critical behavior for D A < D B is governed by a new fixed point. We calculate the critical exponents of the stationary state in an ε expansion, carried out by Wilson renormalization, below the critical dimension d c = 4. We then go on to to obtain the critical initial time behavior at the extinction threshold, both for D A = D B and D A < D B . There is nonuniversal dependence on the initial particle distribution. The case D A > D B remains unsolved.PACS 05.40+j, 05.70.Ln, 82.20.Mj

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of shortrange infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as 1/R d+σ . By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an ε-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection σ = σc > 2.

The persistence exponent θ for the global order parameter, M (t), of a system quenched from the disordered phase to its critical point describes the probability, p(t) ∼ t −θ , that M (t) does not change sign in the time interval t following the quench. We calculate θ to O(ǫ 2 ) for model A of critical dynamics (and to order ǫ for model C) and show that at this order M (t) is a non-Markov process. Consequently, θ is a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Sire [Phys. Rev. Lett. 77, 1420 (1996)].The 'persistence exponent', θ, which characterizes the decay of the probability that a stochastic variable exceeds a threshold value (typically its mean value) throughout a time interval, has attracted a great deal of recent interest [1][2][3][4][5][6][7][8][9][10][11]. One of the most surprising properties of this exponent is that its value is highly non-trivial even in simple systems. For example, θ is irrational for the q > 2 Potts model in one dimension [6] (where the fraction of spins that have not changed their state in the time t after a quench to T = 0 decays as t −θ ) and is apparently not a simple fraction for the diffusion equation [9,10] (where the fraction of space where the diffusion field has always exceeded its mean decays as t −θ ).A recent study of non-equilibrium model A critical dynamics, where a system coarsens at its critical point starting from a disordered initial condition, looked at the probability P (t 1 , t 2 ) that the global magnetization does not change sign during the interval t 1 < t < t 2 [11]. The persistence exponent for this system is defined by P (t 1 , t 2 ) ∼ (t 1 /t 2 ) θ in the limit t 2 /t 1 → ∞. Explicit results were obtained for the 1D Ising model, the n → ∞ limit of the O(n) model, and to order ǫ = 4 − d near dimension d = 4. For these systems it was found that the value of θ was related to the dynamic critical exponent z, the static critical exponent η, and 'nonequilibrium' exponent λ (which describes the decay of the autocorrelation with the initial condition, φ(x, t)φ(x, 0) ∼ t −λ/z ) by the scaling relation θz = λ − d + 1 − η/2. This relation may be derived from the assumption that the dynamics is Markovian, which is indeed the case for all of the cases considered in that paper.From a consideration of the structure of the diagrams which appear at order ǫ 2 (and higher order), however, it was argued that the dynamics of the global order parameter should not be Markovian to all orders, implying that the exponent θ does not obey exactly that 'Markovian scaling relation ' [11]. This means that θ is a new exponent. Monte-Carlo simulations in 2 dimensions indeed suggest weak violation of the Markov scaling relation [11].In this Rapid Communication we present an explicit calculation of the non-Markovian properties of the global order parameter. The nonequilibrium magnetizationmagnetization correlation function is calculated to order ǫ 2 , and this is then used ...