%\documentstyle[aps,preprint,floats,epsf]{revtex} \documentstyle[aps,prl,floats,epsf]{revtex} \begin{document} \advance\textheight by 0.2in \draft % to appear in Phys. Rev. Lett. (1998), cond-mat/9709057 %%%%%%%%%%%%%%% VERSION 16 - 1 - 1998 %%%%%%%%%%%%%%%%%%%%% \twocolumn[\hsize\textwidth\columnwidth\hsize\csname@twocolumnfalse% \endcsname \title{Multicritical Behavior in Coupled Directed Percolation Processes} \author{Uwe C. T\"auber$^{1,*}$, Martin J. Howard$^2$, and Haye Hinrichsen$^3$} \address{$^1$ Department of Physics --- Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP \\ and Linacre College, St. Cross Road, Oxford OX1 3JA, United Kingdom \\ $^2$ CATS, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen \O, Denmark \\ $^3$ Max-Planck-Institut f\"ur Physik komplexer Systeme, N\"othnitzer Stra\ss e 38, D-01187 Dresden, Germany \\ } \date{\today, submitted to Phys. Rev. Lett., OUTP-97-24-S} \maketitle \begin{abstract} We study a hierarchy of directed percolation (DP) processes for particle species $A, B, \ldots$, unidirectionally coupled via the reactions $A \to B, \ldots$. When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents $\beta^{(k)}$ which are markedly reduced at each hierarchy level $k \geq 2$. We compute the fluctuation corrections to $\beta^{(2)}$ to $O(\epsilon = 4-d)$ using field-theoretic renormalization group techniques. Monte Carlo simulations are employed to determine the new exponents in dimensions $d \leq 3$. \end{abstract} %\pacs{PACS numbers: 64.60.Ak, 05.40.+j, 82.20.-w.} \pacs{PACS numbers: 64.60.Ak, 05.40.+j, 82.20.-w.}] % PACS: 64.60.Ak ---- Renormalization-group, fractal, and percolation % studies of phase transitions % 05.40.+j ---- Fluctuation phenomena, random processes, and % Brownian motion % 82.20.-w ---- Chemical kinetics Nonequilibrium critical phenomena have been the focus of intense theoretical studies in recent years, prominent examples being phase transitions in driven diffusive systems \cite{dridif}, kinetic Ising models \cite{kinism}, nonequilibrium growth models \cite{gromod}, diffusion-limited chemical reactions \cite{chemrc,jcardy}, percolation-like processes \cite{percol}, and generally in lattice models or cellular automata. The concept of only a few distinct universality classes which determine the critical exponents has been remarkably successful for the description of static and even dynamic critical phenomena near equilibrium phase transitions \cite{crtdyn}. However, it is still an unresolved issue whether a similarly small number of universality classes can be identified in nonequilibrium situations, which potentially contain much richer behavior. In addition to the dynamic universality classes listed in Ref.~\cite{crtdyn}, some notable new candidates have emerged in quite different circumstances, for example, the roughening transition in the KPZ equation \cite{kapezt}, and the critical point of directed percolation (DP) \cite{dirper}, as described by Reggeon field theory \cite{regfth}. In fact, the DP universality class appears to cover the majority of phase transitions from non-trivial active into absorbing states where the order parameter noise vanishes. However one striking exception to this rule occurs when the relevant dynamic processes are constrained by an additional local ``parity'' symmetry, in which case the universality class is that of branching and annihilating random walks with an even number of offspring \cite{evbarw,cartau}. This work is originally motivated by recent studies of a nonequilibrium growth model for adsorption and desorption of particles which displays a roughening transition in $d = 1$ \cite{hinric}. The key feature is that desorption may take place only at the edge of an existing plateau so that particles cannot be desorbed from a completed layer. Because of this property the dynamic processes at a given height level decouple from all higher levels, i.e. the processes at different heights are unidirectionally coupled. This growth model can in fact be related to a reaction-diffusion problem for a hierarchy of particle species $A, B, \ldots$, which correspond to different height levels, respectively. Each of these species is subject to the prototypical DP processes (see Refs.~\cite{regfth,cartau}) % \begin{equation} A \to A + A \ , \quad A + A \to A \ , \quad A \to \emptyset \ , \label{dpproc} \end{equation} % with reaction rates $\sigma_A$, $\lambda_A$, and $\mu_A$, respectively (and accordingly for particles $B, \ldots$). However, there is also a coupling from each previous hierarchy level to the next one via random particle transmutations % \begin{equation} A \to B \ , \quad B \to C \ , \quad \ldots \label{coupdp} \end{equation} % with rates $\mu_{AB}, \mu_{BC}, \ldots$, but {\em no} feedback mechanism from the $B$ to the $A$ species etc. For simplicity, we shall assume that the diffusion constant $D$ for {\em all} particle species $A, B, \ldots$ is identical. When $\mu_{AB} = \mu_{BC} = \ldots = 0$, the system decouples, and for the $i$-th species there is a critical point separating an active from an absorbing state, where essentially the branching rate $\sigma_i$ and decay rate $\mu_i$ balance one another. At this point there are {\em three} independent critical exponents governing ({\it i}) the divergence of the correlation length $\xi \propto |r_i|^{-\nu_\perp}$ (where $r_i \propto \mu_i - \sigma_i$ denotes the deviation from the critical point), ({\it ii}) the critical slowing down of characteristic time scales $t_c \propto \xi^z \propto |r_i|^{-\nu_\parallel}$, with $\nu_\parallel = z \nu_\perp$, and ({\it iii}) the value of the asymptotic particle density in the active phase, $n_i(t \to \infty) \propto |r_i|^\beta$. Alternatively, this third independent exponent may be swapped for $\eta_\perp$, which characterizes how the equal-time pair correlation function decays at the critical point $r_i = 0$, $G(|{\bf x}|) \propto 1/|{\bf x}|^{d + z - 2 + \eta_\perp}$ [see Eq.~(\ref{dpbeta}) below]. These exponents should be those of the DP universality class, with upper critical (spatial) dimension $d_c = 4$. Numerical simulations in $d = 1$ have confirmed this, and furthermore revealed that $\nu_\perp \approx 1.1$ and $z \approx 1.6$ remain unchanged at {\em all} hierarchy levels even when the rates $\mu_{AB} \ldots$ are switched on. However, while $\beta_A = \beta^{(1)} \approx 0.27$ as in DP for the primary $A$ particles, the particle density exponents for secondary and higher hierarchy levels are considerably reduced to $\beta_B = \beta^{(2)} \approx 0.11$, and $\beta^{(3)} \approx 0.04$, {\em provided} the critical points for the processes on the different levels coincide \cite{hinric}. In the following, we shall mainly study the two-level hierarchy of DP processes (\ref{dpproc}) for $A$ and $B$ particles coupled by the reaction $A \to B$. We explain the reduced value of $\beta^{(2)}$ as a consequence of the {\em multicritical behavior} that emerges when $r_A = r_B = r \to 0$. It appears then rather likely that similar multicritical behavior should emerge in more general contexts than the specific growth model and diffusion-limited chemical reactions (\ref{dpproc}), (\ref{coupdp}); namely whenever DP-like processes are coupled unidirectionally {\em without} feedback. A qualitatively correct description is obtained by analyzing the corresponding mean-field rate equations, which yield $\beta^{(k)} = 1/2^{k-1}$ at the multicritical point on the $k$-th hierarchy level. With the aid of the renormalization group (RG), we then identify an additional {\em independent} crossover exponent $\phi$ related to the relevant scaling field ($\mu_{AB} / D$) of the two-level hierarchy multicritical point. A scaling relation will be derived expressing $\beta^{(2)}$ in terms of $\phi$ and the conventional DP exponent $\beta = \beta^{(1)}$. Using a field-theoretic representation (see Refs.~\cite{masfth,redfth}) based on the master equation for this stochastic process, we furthermore compute the fluctuation corrections to $\phi$ and $\beta^{(2)}$ to first order in $\epsilon = 4 - d$ \cite{yadin}. Finally, these exponents are determined accurately in $d \leq 3$ dimensions by means of Monte Carlo simulations. \begin{table} \begin{tabular}{|c||l|l|l|l|} & $d=1$ & $d=2$ & $d=3$ & $d=4-\epsilon$ \\\hline $\beta^{(1)}$ & $0.271(10)$ & $0.57(5)$ & $0.78(7)$ & $1-\epsilon/6+ O(\epsilon^2) $\\ $\beta^{(2)}$ & $0.108(10)$ & $0.30(4)$ & $0.35(6)$ & $1/2 - \epsilon / 6 + O(\epsilon^2) $ \\ $\beta^{(3)}$ & $0.038(8)$ & $0.13(3)$ & $0.15(4)$ & $1/4 - \epsilon / 6 + O(\epsilon^2) $ \\ \end{tabular} \caption{Results for the particle density exponents $\beta^{(k)}$ obtained from Monte Carlo simulations and RG calculations.} \end{table} We start by writing down the mean-field rate equations for the average local densities $n_A({\bf x},t)$ and $n_B({\bf x},t)$ of the $A$ and $B$ particles, both subject to the reactions (\ref{dpproc}), and coupled via the random particle transmutation $A \to B$, % \begin{eqnarray} \partial_t n_A &&= D (\nabla^2 - r_A) n_A - \lambda_A n_A^2 \ , \label{mfreqa} \\ \partial_t n_B &&= D (\nabla^2 - r_B) n_B - \lambda_B n_B^2 + \mu_{AB} n_A \ , \label{mfreqb} \end{eqnarray} % where $r_A = (\mu_A + \mu_{AB} - \sigma_A) / D$ and $r_B = (\mu_B - \sigma_B) / D$. Considering first the equation for $n_A$, it is clear that for $r_A > 0$ the only stationary state is $n_A = 0$ (inactive phase). On the other hand, if $r_A < 0$ the density will asymptotically approach $n_A^\infty = D (-r_A) / \lambda_A > 0$ (active state), and hence $\beta^{(1)} = 1$ in this approximation. From the diffusive character of Eq.~(\ref{mfreqa}), and also from the functional form of its linear term, it follows that $z = 2$ and $\nu_\perp = 1/2$ in mean-field. These last two results remain intact for the $B$ particles as well, even in the presence of particle influx via the reaction $A \to B$. However, the possible stationary states for $n_B$ actually depend on the value of $r_A$ as well as on $r_B$, if $\mu_{AB} > 0$. If $r_A > 0$, and therefore $n_A(t \to \infty) = 0$, Eq.~(\ref{mfreqb}) reduces to a mean-field DP process with critical point $r_B = 0$ and $\beta = 1$. On the other hand, in the $A$-particle active state, $r_A < 0$, the stationary solution of (\ref{mfreqb}) becomes $n_B^\infty = [(D r_B / 2 \lambda_B)^2 + \mu_{AB} n_A^\infty / \lambda_B]^{1/2} - D r_B / 2 \lambda_B$; i.e., there is merely a crossover from a low-density to a high-density regime in the vicinity of $r_B = 0$. An interesting situation arises when $r_A = r_B = r$, as in this case the second term in the square brackets dominates for $|r| \to 0$, % \begin{equation} n_B^\infty = [D \mu_{AB} (-r)/ \lambda_A \lambda_B]^{1/2} + O(|r|) \ . \label{mfdens} \end{equation} % Hence $\beta^{(2)} = 1/2$ instead of the mean-field DP value. More generally, this multicritical behavior arises in the regime $(D r_B / 2 \lambda_B)^2 \ll D (-r_A) \mu_{AB} / \lambda_A \lambda_B$ for $r_A \uparrow 0$. On the other hand, for $r_B > 0$ and $(D r_B / 2 \lambda_B)^2 \gg D (-r_A) \mu_{AB} / \lambda_A \lambda_B$ one finds $n_B^\infty = (-r_A) \mu_{AB} / r_B \lambda_A$, and thus one expects DP transitions for the $B$ species {\em both} as $r_B \to 0$ (with $r_A > 0$) {\em and} $r_A \to 0$ ($r_B > 0$) \cite{future}. Notice that there are three critical lines, namely ($r_A = 0$, $r_B > 0$), ($r_A = 0$, $r_B < 0$), and ($r_A > 0$, $r_B = 0$), two of which are critical lines for the $B$ particles, meeting at the point $r_A = r_B = 0$. This special point can therefore be interpreted as a multicritical point, with the mean-field order parameter exponent halved as compared to the ordinary critical point \cite{eqtric}. Clearly the generalization of Eqs.~(\ref{mfreqa}), (\ref{mfreqb}) to $k$ hierarchy levels leads to $\beta^{(k)} = 1/2^{k-1}$ at the multicritical point $r_A = r_B = \ldots = 0$. For $d < d_c = 4$, one expects that these mean-field values will be strongly modified by fluctuation effects. We have performed Monte Carlo simulations in $d \leq 3$ dimensions in order to determine the survival probability exponent $\beta^{(k)}/\nu_\parallel^{(k)}$ up to the third hierarchy level. The DP processes (\ref{dpproc}) are realized by a cellular automaton which evolves by parallel updates in which a site at time $t+1$ becomes active with probability $p$ provided that the same site, or at least one of its nearest neighbors, was active at time $t$. The coupling (\ref{coupdp}) between levels is incorporated by the rule that active sites at a given level simultaneously impose active sites on all higher levels. We perform dynamic Monte Carlo simulations \cite{grassb} where one measures the temporal evolution of the system at criticality starting from a single seed (a single $A$ particle). Assuming that $\nu_\parallel^{(k)} = \nu_\parallel$ is identical on all levels $k$, we can use the survival probability to determine the density exponents, giving in $d = 1$ $\beta^{(1)} = 0.271(10)$ as in DP \cite{jensen}, $\beta^{(2)} = 0.108(10)$, and $\beta^{(3)} = 0.038(8)$. These results are consistent with direct estimations of $\beta^{(k)}$ obtained in off-critical simulations \cite{hinric}. The mean-square spreading exponents yield the exponent $z$, with the results $z^{(1)} = 1.57(2)$ (DP), $z^{(2)} = 1.58(2)$ and $z^{(3)} = 1.59(2)$, which confirms that these exponents are identical on all levels. Similar results are obtained in $d = 2$, $d = 3$ (see Table I). In order to better understand the very low values found for $\beta^{(k)}$ in the simulations, we clearly have to take fluctuation effects into account. In low dimensions, where reaction noise in the processes (\ref{dpproc}), (\ref{coupdp}) is highly important, a straightforward Langevin-type description of the noise can be dangerous, because of the highly non-trivial form of the noise correlations. Instead, it is useful to start from the corresponding master equation, and utilize a standard formalism involving a second-quantized bosonic operator representation to derive an effective field theory directly \cite{masfth,redfth}. It is important to note that apart from the continuum limit, this procedure is {\em exact} and requires no additional assumptions regarding the precise form of the noise. In the present case, the field theory describing the two-level coupled reactions in $d$ dimensions is defined by the action (we omit terms related to the initial state) % \begin{eqnarray} S &&= \int \! d^dx \int \! dt \Bigl\{ {\bar a} \left[ \partial_t + D (r_A - \nabla^2) \right] a - \sigma_A {\bar a}^2 a \nonumber \\ &&+ \lambda_A \left( {\bar a} a^2 + {\bar a}^2 a^2 \right) - \mu_{AB} {\bar b} a \\ \label{fthact} &&+ {\bar b} \left[ \partial_t + D (r_B - \nabla^2) \right] b - \sigma_B {\bar b}^2 b + \lambda_B \left( {\bar b} b^2 + {\bar b}^2 b^2 \right) \Bigr\} \ . \nonumber \end{eqnarray} % We remark that $\langle a \rangle = n_A$ and $\langle b \rangle = n_B$, whereas the fields ${\bar a}, {\bar b}$ have no direct physical interpretation. It is now convenient to rescale the fields according to ${\bar a} = (\lambda_A / \sigma_A)^{1/2} {\bar \psi}_0$, $a = (\sigma_A / \lambda_A)^{1/2} \psi_0$, ${\bar b} = (\lambda_B / \sigma_B)^{1/2} {\bar \varphi}_0$, $b = (\sigma_B / \lambda_B)^{1/2} \varphi_0$, and also define new couplings $u_0 = 2 (\sigma_A \lambda_A)^{1/2}$, $u_0' = 2 (\sigma_B \lambda_B)^{1/2}$, as well as $\mu_0 = \mu_{AB} (\sigma_A \lambda_B / \sigma_B \lambda_A)^{1/2}$ (henceforth, the subscript ``0'' denotes unrenormalized quantities). If we introduce a length scale $\kappa^{-1}$ and correspondingly measure times in units of $\kappa^{-2}$ (i.e., $[D_0] = \kappa^0$), we find that the new fields have scaling dimension $\kappa^{d/2}$, while $[r_A] = [r_B] = [\mu_0] = \kappa^2$, which are thus relevant perturbations in the RG sense. On the other hand, $[u_0] = [u_0'] = \kappa^{2-d/2}$, and the corresponding DP nonlinearities become marginal in $d_c = 4$ dimensions, as expected \cite{regfth}. It is important to note, however, that $[\lambda_A] = [\lambda_B] = \kappa^{2-d}$, and hence these couplings are {\em irrelevant} as compared to $u_0$ and $u_0'$, and may be omitted in the effective action (see Ref.~\cite{cartau}), which consequently reads % \begin{eqnarray} S_{\rm eff} &&= \int \! d^dx \int \! dt \Bigl\{ {\bar \psi}_0 \left[ \partial_t + D_0 (r_A - \nabla^2) \right] \psi_0 \nonumber \\ &&- {u_0 \over 2} \left( {\bar \psi}_0^2 \psi_0 - {\bar \psi}_0 \psi_0^2 \right) - \mu_0 {\bar \varphi}_0 \psi_0 \label{effact} \\ &&+ {\bar \varphi}_0 \left[ \partial_t + D_0 (r_B - \nabla^2) \right] \varphi_0 - {u_0' \over 2} \left( {\bar \varphi}_0^2 \varphi_0 - {\bar \varphi}_0 \varphi_0^2 \right) \Bigr\} \ . \nonumber \end{eqnarray} % The saddle-point equations corresponding to this action are precisely the mean-field rate equations (\ref{mfreqa}), (\ref{mfreqb}) but with $\psi_0$, $\varphi_0$ instead of $n_A, n_B$, and with $\mu_{AB}$, $\lambda_A$, and $\lambda_B$ replaced by $\mu$, $u_0/2$, and $u_0'/2$. We remark that the harmonic part of this action can be diagonalized {\em only} when $r_A \not= r_B$, leading to new cubic vertices mixing the fields $\psi_0$ and $\varphi_0$ [compare Eq.~(\ref{addver}) below]. However, these additional vertices have no influence on the behavior of the $A$ and $B$ species \cite{jansen}, and the diagonalized theory leads to the anticipated DP critical behavior at both critical lines ($r_A = 0$, $r_B \not= 0$) and ($r_A > 0$, $r_B = 0$) \cite{future}. Here we are predominantly interested in the multicritical point $r_A = r_B = r_0 \to 0$, and thus we must work with the non-diagonal action (\ref{effact}). Furthermore, due to the inclusion of the relevant scaling field $\mu_0$, a number of ``mixed'' cubic vertices are additionally generated at the tree level, all of which have scaling dimension $\kappa^{2-d/2}$. These vertices must be taken into account separately, and hence we need to replace (\ref{effact}) with $S_{\rm mc} = S_{\rm eff} + \Delta S$, where % \begin{eqnarray} \Delta S = \int \! d^dx \int \! dt \Bigl[ &&- s_0 {\bar \varphi}_0 {\bar \psi}_0 \psi_0 - {s_0' \over 2} {\bar \varphi}_0^2 \psi_0 \nonumber \\ &&+ {{\tilde s_0} \over 2} {\bar \varphi}_0 \psi_0^2 + {\tilde s}_0' {\bar \varphi}_0 \varphi_0 \psi_0 \Bigr] \ . \label{addver} \end{eqnarray} % It is now a straightforward task to compute the renormalized couplings and the RG fixed points to one-loop order. The DP nonlinearities in (\ref{effact}) remain unaffected by the ``mixed'' vertices (\ref{addver}), and the stable nontrivial fixed points for the associated dimensionless renormalized couplings $u = u_0 A_d^{1/2} \kappa^{(d-4)/2}$ and $u' = u_0' A_d^{1/2} \kappa^{(d-4)/2}$, where $A_d = \Gamma(3 - d/2) / 2^{d-1} \pi^{d/2}$, are % \begin{equation} [(u/D)^*]^2 = [(u'/D)^*]^2 = 4 \epsilon/3 + O(\epsilon^2) \ . \label{dpfxpt} \end{equation} % Hence the critical exponents $\eta_\perp = - \epsilon / 12 + O(\epsilon^2)$, $\nu_\perp^{-1} = 2 - \epsilon / 4 + O(\epsilon^2)$, and $z = 2 - \epsilon / 12 + O(\epsilon^2)$ remain those of the DP universality class at {\it all} hierarchy levels, whereas the exponent $\beta^{(k)}$ is unchanged only at the first level % \begin{equation} \beta^{(1)} = {\nu_\perp \over 2} (d + z - 2 + \eta_\perp) = 1 - \epsilon / 6 + O(\epsilon^2) \ . \label{dpbeta} \end{equation} % For the similarly defined renormalized ``mixed'' cubic vertices, one finds two RG {\em fixed lines}. The first is given by % \begin{eqnarray} &&(s/D)^* = - ({\tilde s}'/D)^* , \, ({\tilde s}/D)^* - (s'/D)^* = 2 (s/D)^* , \label{fline1} \\ &&\qquad [(s/D)^*]^2 = 2 (\epsilon/3)^{1/2} [(s/D)^* + (s'/D)^*] \ , \label{connec} \end{eqnarray} % with stability matrix eigenvalues $0, 0, -\epsilon/3, -\epsilon/3$, which imply that this fixed line (including the Gaussian fixed point for $\Delta S$) is unstable for $d < 4$. The second fixed line, % \begin{equation} (s'/D)^* = ({\tilde s}/D)^* , \, (s/D)^* + ({\tilde s}'/D)^* = 2 (\epsilon/3)^{1/2} , \label{fline2} \end{equation} % with again Eq.~(\ref{connec}), turns out to be {\em stable} as its stability matrix has eigenvalues $0, \epsilon/3, \epsilon/3, 4 \epsilon/3$ \cite{future}. We remark that both fixed lines (\ref{fline1}) and (\ref{fline2}) with (\ref{connec}) satisfy the condition $(\epsilon/3)^{1/2} [(s'/D)^* + ({\tilde s}/D)^*] = - (s/D)^* ({\tilde s}'/D)^*$, which ensures the cancellation of strongly singular (UV-divergent in $d = 2$) diagrams for the renormalization of $\mu_0$. The RG eigenvalue of $\mu$ at these fixed lines then becomes $y_\mu = 2 + \epsilon / 6$, and $y_\mu = 2 - \epsilon / 6$, respectively. In principle a product of quartic vertices and $\mu_0$ might enter the renormalizations of the three-point functions as well. However, we have checked that these additional couplings all have negative RG eigenvalues and are therefore irrelevant \cite{future}. We next define the crossover exponent $\phi$ related to the new scaling field $\mu / D$, as it appears in the general scaling form for the $B$--species ``density'' field, % \begin{equation} \varphi(|r|,\mu/D) = |r|^{\beta^{(1)}} \, {\hat \varphi}(|r|^{-\phi} \, \mu / D) \label{densca} \end{equation} % and identify % \begin{equation} \phi = \nu_\perp (y_\mu + z - 2) = 1 + O(\epsilon^2) \label{1lpchi} \end{equation} % at the stable fixed line. Finally, we relate the above exponent $\phi$ to the density exponent $\beta^{(2)}$ on the second hierarchy level. We first use the RG flow equations for the scale-dependent renormalized parameters, and then match to our earlier mean-field result (\ref{mfdens}) for the density in the active phase in the vicinity of the multicritical point. In terms of the rescaled fields, we found $\varphi_0^\infty = (2\mu_0 \psi_0^\infty / u_0')^{1/2}$ for $r_0 \to 0$. Essentially, we now have to replace the bare parameters here by their flowing renormalized counterparts, when inverse length scales are rescaled according to $\kappa \to \kappa \ell$. To that end, we first note that the fields $\psi$ and $\varphi$ scale as $\ell^{(d + z - 2 + \eta_\perp)/2}$ [which, with the matching condition $\ell = (-r)^{\nu_\perp}$, actually implies the scaling relation cited in Eq.~(\ref{dpbeta})]. Furthermore, the $O(\epsilon)$ fixed points (\ref{dpfxpt}) require $u^{(')}(\ell) / D(\ell) \to {\rm const.}$, and according to our definitions, we have $r(\ell) \to r \ell^{-1/\nu_\perp}$ and $\mu(\ell) / D(\ell) \propto \ell^{- \phi / \nu_\perp}$. Upon identifying $\ell = (-r)^{\nu_\perp}$, we finally arrive at % \begin{equation} \beta^{(2)} = \beta^{(1)} - \phi / 2 = 1/2 - \epsilon / 6 + O(\epsilon^2) \ , \label{tpbeta} \end{equation} % using our above results \cite{yadin}. Equivalently, this follows by demanding that the scaling function in Eq.~(\ref{densca}) behave as ${\hat \varphi}(x) \to x^{1/2}$ for $x \to \infty$, as prescribed by mean-field theory (\ref{mfdens}). The $O(\epsilon)$ result (\ref{tpbeta}) constitutes a sizable {\em downward} renormalization through fluctuation effects, as is also evident from the simulation data in Table I. These can be explained with the values $\phi \approx 0.33$, $0.54$, and $0.86$ in $d = 1$, $2$, and $3$, respectively. In principle, these considerations are readily generalized to further hierarchy levels \cite{future}. For example, although a combination of processes $A \to B$ and $B \to C$ will generate the reaction $A \to C$ as well, one finds that the additional relevant scaling field $\mu_{AC}$ does not enter the leading contribution to the $C$ density near the multicritical point $r_A = r_B = r_C = r \to 0$. Hence $n_C \approx [D \mu_{AB} \mu_{BC}^2 (-r) / \lambda_A \lambda_B \lambda_C^2]^{1/4}$, implying that no additional independent exponents need to be introduced. We then find $\beta^{(3)} = \beta^{(1)} - 3 \phi / 4 = 1/4 - \epsilon / 6 + O(\epsilon^2)$. In summary, we have studied a hierarchy of {\em unidirectionally coupled} DP processes (no feedback), as defined by the diffusion-limited reactions (\ref{dpproc}), (\ref{coupdp}). Already within mean-field theory, we have identified a purely {\em dynamic multicritical point}, which occurs when the DP critical points at each level coincide. Using a field-theoretic representation, we have evaluated to $O(\epsilon)$ the crossover exponent $\phi$ related to the relevant scaling field $\mu_{AB} / D$. We then derived a scaling relation which allowed us to determine $\beta^{(2)}$. In accord with our Monte Carlo simulation results, fluctuation effects lead to a strong {\em downward} renormalization of the density exponents as compared with the mean-field values $\beta^{(k)} = 1/2^{k-1}$. We emphasize that the existence of such a multicritical regime is to be expected generically whenever processes of the ubiquitous DP universality class are coupled unidirectionally. We benefited from discussions with J.L.~Cardy, M.R.~Evans, Y.Y.~Goldschmidt, and D.~Mukamel. U.C.T. acknowledges support from the European Commission, contract No.~ERB FMBI-CT96-1189, and from the Deutsche Forschungsgemeinschaft, Gz.~Ta 177 / 2. $^*$ Present address: Institut f\"ur Theoretische Physik, Physik-Department der TU M\"unchen, James-Franck-Stra\ss e, D-85747 Garching, Germany. %\vspace{-.1in} \begin{thebibliography}{99} %\vspace{-.2in} \bibitem{dridif} B.~Schmittmann and R.K.P.~Zia, in {\em Phase Transitions and Critical Phenomena}, eds. C.~Domb and J.L.~Lebowitz (Academic Press, New York, 1996). \bibitem{kinism} M.~Droz, Z.~R\'acz, and J.~Schmidt, Phys. Rev. A {\bf 39}, 2141 (1989). \bibitem{gromod} A.L.~Barab\'asi and H.E.~Stanley, {\em Fractal Concepts in Surface Growth} (Cambridge University Press, Cambridge, 1995); J.~Krug, Adv. Phys. {\bf 46}, 139 (1997). \bibitem{chemrc} V.~Kuzovkov and E.~Kotomin, Rep. Prog. Phys. {\bf 51}, 1479 (1988). \bibitem{jcardy} J.L.~Cardy, in {\em Proceedings of mathematical beauty of physics}, ed. J.-B.~Zuber, Adv. Ser. in Math. Phys., vol. 24, to appear (report cond-mat/9607163). \bibitem{percol}{\em Percolation Structures and Processes}, ed. G.~Deutscher, R.~Zallen, and J.~Adler, Ann. Isr. Phys. Soc. {\bf 5} (Adam Hilger, Bristol, 1983). \bibitem{crtdyn} P.C.~Hohenberg and B.I.~Halperin, Rev. Mod. Phys. {\bf 49}, 435 (1977). \bibitem{kapezt} M.~Kardar, G.~Parisi, and Y.-C.~Zhang, Phys. Rev. Lett. {\bf 56}, 889 (1986); for reviews, see Ref.~\cite{gromod} and T.~Halpin-Healy and Y.-C.~Zhang, Phys.\ Rep.\ {\bf 254}, 215 (1995). \bibitem{dirper} For a review, see W.~Kinzel, in Ref.~\cite{percol}, p.~425. \bibitem{regfth} P.~Grassberger and K.~Sundermeyer, Phys. Lett. {\bf 77 B}, 220 (1978). J.L.~Cardy and R.L.~Sugar, J. Phys. A {\bf 13}, L423 (1980); H.K.~Janssen, Z. Phys. B {\bf 42}, 151 (1981). \bibitem{evbarw} P.~Grassberger, F.~Krause, and T.~van~der~Twer, J. Phys. A {\bf 17} L105 (1984); H.~Takayasu and A.Yu.~Tretyakov, Phys. Rev. Lett. {\bf 68}, 3060 (1992); N.~Menyh\'ard and G.~\'Odor, J. Phys. A {\bf 28}, 4505 (1995); K.E.~Bassler and D.A.~Browne, Phys. Rev. Lett. {\bf 77}, 4094 (1996); H.~Hinrichsen, Phys. Rev. E {\bf 55}, 219 (1997). \bibitem{cartau} J.L.~Cardy and U.C.~T\"auber, Phys. Rev. Lett. {\bf 77}, 4780 (1996); (report cond-mat/9704160). \bibitem{hinric} U.~Alon, M.R.~Evans, H.~Hinrichsen, and D.~Mukamel, Phys. Rev. Lett. {\bf 76}, 2746 (1996); (unpublished). \bibitem{masfth} M.~Doi, J. Phys. A {\bf 9}, 1479 (1976); P.~Grassberger and P.~Scheunert, Fortschr. Phys. {\bf 28}, 547 (1980); L.~Peliti, J. Phys. (Paris) {\bf 46}, 1469 (1984); B.P.~Lee, J. Phys. A {\bf 27}, 2633 (1994). \bibitem{redfth} B.P.~Lee and J.L.~Cardy, J. Stat. Phys. {\bf 80}, 971 (1995); M.J.~Howard and J.L.~Cardy, J. Phys. A {\bf 28}, 3599 (1995); P.-A.~Rey and M.~Droz, J. Phys. A {\bf 30}, 1101 (1997); M.J.~Howard and U.C.~T\"auber, J. Phys. A {\bf 30} 7721 (1997). \bibitem{yadin} The one-loop results were also independently derived by Y.Y. Goldschmidt (unpublished, private communication). \bibitem{future} Further details, as well as some subtleties of the theory in the active phase (Y.Y. Goldschmidt, private communication) will be presented elsewhere: U.C.~T\"auber, M.J.~Howard, and H.~Hinrichsen (unpublished). \bibitem{eqtric} This is remarkably similar to the equilibrium tricritical behavior in the ($r \Phi^2 + u \Phi^4 + v \Phi^6$)-model, where $\beta = 1/2$ for $u > 0$ and $\beta = 1/4$ for $u = 0$ in the Landau approximation. Notice, however, that the mechanism leading to the reduction of the exponent $\beta$ is quite different here. \bibitem{grassb} P. Grassberger and A. de la Torre, Ann. Phys. (N.Y.) {\bf 122}, 373 (1979). \bibitem{jensen} I.~Jensen and A.J.~Guttmann, J. Phys. A {\bf 28}, 4813 (1995); I.~Jensen, Phys. Rev. Lett. {\bf 77}, 4988 (1996). \bibitem{jansen} H.K.~Janssen, Phys. Rev. Lett. {\bf 78}, 2890 (1997). \end{thebibliography} \end{document}