Directed Percolation Transition Research Articles

Triviality of quantum trajectories close to a directed percolation transition

We study quantum circuits consisting of unitary gates, projective measurements, and control operations that steer the system toward a pure absorbing state. Two types of phase transition occur as the rate of these control operations is increased: a measurement-induced entanglement transition, and a directed percolation transition into the absorbing state (taken here to be a product state). In this work, we show analytically that these transitions are generically distinct, with the quantum trajectories becoming disentangled before the absorbing state transition is reached, and we analyze their critical properties. We introduce a simple class of models where the measurements in each quantum trajectory define an effective tensor network (ETN)---a subgraph of the initial spacetime graph where nontrivial time evolution takes place. By analyzing the entanglement properties of the ETN, we show that the entanglement and absorbing-state transitions coincide only in the limit of the infinite local Hilbert-space dimension. Focusing on a Clifford model which allows numerical simulations for large system sizes, we verify our predictions and study the finite-size crossover between the two transitions at large local Hilbert space dimension. We give evidence that the entanglement transition is governed by the same fixed point as in hybrid circuits without feedback.

Structural and dynamical properties of dilute gel networks in colloid-polymer mixtures.

The competition of short-ranged depletion attraction and long-ranged repulsion between colloidal particles in colloid-polymer mixtures leads to the formation of heterogeneous gel-like structures. Our special focus will be on the states where the colloids arrange in thin strands that span the whole system and that we will refer to as dilute gel networks. These states occur at low packing fractions for attractions that are stronger than those at both the binodal line of the equilibrium gas-liquid phase separation and the directed percolation transition line. By using Brownian dynamics simulations, we explore the formation, structure, and aging dynamics of dilute gel networks. The essential connections in a dilute gel network are determined by constructing reduced networks. We compare the observed properties to those of clumpy gels or cluster fluids. Our results demonstrate that both the structure and the (often slow) dynamics of the stable or meta-stable heterogeneous states in colloid-polymer mixtures possess distinct features on various length and time scales and thus are richly diverse.

Novel transition to fully absorbing state without long-range spatial order in directed percolation class

We study coupled Gauss maps in one dimension and observe a transition to band periodic state with 2 bands. This is a periodic state with period-2 in a coarse-grained sense. This state does not show any long-range order in space. We compute two different order parameters to quantify the transition a) Flipping rate F(t) which measures departures from period-2 and b) Persistence P(t) which quantifies the loss of memory of initial conditions. At the critical point, F(t) shows a power-law decay with exponent 0.158 which is close to 1-D directed percolation (DP) transition. The persistence exponent at the critical point is found to be 1.51 which matches with several models in 1-D DP class. We also study the finite-size scaling and off-critical scaling to estimate other exponents z and ν∥. We observe excellent scaling for both F(t) as well as P(t) and the exponents obtained are clearly in DP class. We believe that DP transition could be observed in systems where activity goes to zero even if the spatial profile could be inhomogeneous and lacking any long-range order.

Bacterial range expansions on a growing front: Roughness, fixation, and directed percolation.

Directed percolation (DP) is a classic model for nonequilibrium phase transitions into a single absorbing state (fixation). It has been extensively studied by analytical and numerical techniques in diverse contexts. Recently, DP has appeared as a generic model for the evolutionary and ecological dynamics of competing bacterial populations. Range expansion-the stochastic reproduction of bacteria competing for space to be occupied by their progeny-leads to a fluctuating and rough growth front, which is known from experiment and simulation to affect the underlying critical behavior of the DP transition. In this work, we employ symmetry arguments to construct a pair of nonlinear stochastic partial differential equations describing the coevolution of surface roughness with the composition field of DP. Macroscopic manifestations (phenomenology) of these equations on growth patterns and genealogical tracks of range expansion are discussed; followed by a renormalization group analysis of possible scaling behaviors at the DP transition.

Sensitivity of directed networks to the addition and pruning of edges and vertices.

Directed networks have various topologically different extensive components, in contrast to a single giant component in undirected networks. We study the sensitivity (response) of the sizes of these extensive components in directed complex networks to the addition and pruning of edges and vertices. We introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the Caenorhabditis elegans neural network, directed Erdős-Rényi graphs, and others. We reveal a nonmonotonic dependence of the sensitivity to random pruning of edges or vertices in the case of C. elegans and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.

Multiple phase transitions in an agent-based evolutionary model with neutral fitness.

Null models are crucial for understanding evolutionary processes such as speciation and adaptive radiation. We analyse an agent-based null model, considering a case without selection—neutral evolution—in which organisms are defined only by phenotype. Universal dynamics has previously been demonstrated in a related model on a neutral fitness landscape, showing that this system belongs to the directed percolation (DP) universality class. The traditional null condition of neutral fitness (where fitness is defined as the number of offspring each organism produces) is extended here to include equal probability of death among organisms. We identify two types of phase transition: (i) a non-equilibrium DP transition through generational time (i.e. survival), and (ii) an equilibrium ordinary percolation transition through the phenotype space (based on links between mating organisms). Owing to the dynamical rules of the DP reaction–diffusion process, organisms can only sparsely fill the phenotype space, resulting in significant phenotypic diversity within a cluster of mating organisms. This highlights the necessity of understanding hierarchical evolutionary relationships, rather than merely developing taxonomies based on phenotypic similarity, in order to develop models that can explain phylogenetic patterns found in the fossil record or to make hypotheses for the incomplete fossil record of deep time.

Yielding of glass under shear: A directed percolation transition precedes shear-band formation.

Under external mechanical loading, glassy materials, ranging from soft matter systems to metallic alloys, often respond via formation of inhomogeneous flow patterns, during yielding. These inhomogeneities can be precursors to catastrophic failure, implying that a better understanding of their underlying mechanisms could lead to the design of smarter materials. Here, extensive molecular dynamics simulations are used to reveal the emergence of heterogeneous dynamics in a binary Lennard-Jones glass, subjected to a constant strain rate. At a critical strain, this system exhibits for all considered strain rates a transition towards the formation of a percolating cluster of mobile regions. We give evidence that this transition belongs to the universality class of directed percolation. Only at low shear rates, the percolating cluster subsequently evolves into a transient (but long-lived) shear band with a diffusive growth of its width. Finally, the steady state with a homogeneous flow pattern is reached. In the steady state, percolation transitions also do occur constantly, albeit over smaller strain intervals, to maintain the stationary plastic flow in the system.

Message passing theory for percolation models on multiplex networks with link overlap.

Multiplex networks describe a large variety of complex systems, including infrastructures, transportation networks, and biological systems. Most of these networks feature a significant link overlap. It is therefore of particular importance to characterize the mutually connected giant component in these networks. Here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers M. Specifically we propose and compare two message passing algorithms that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap. The first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation. The second algorithm describes the emergence of the mutually connected giant component, that is the percolation transition, but does not preserve the epidemic spreading interpretation. We obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases. We demonstrate that for the same multiplex network structure, in which the directed percolation transition has nontrivial tricritical points, the percolation transition has a discontinuous phase transition, with the exception of the trivial case in which all the layers completely overlap.

Absorbing phase transition in energy exchange models

We study energy exchange models with dissipation (λ) and noise (of amplitude σ) and show that in presence of a threshold these models undergo an absorbing phase transition when either dissipation or noise strength or both are varied. Using Monte Carlo simulations we find that the behaviour along the critical line, which separates the active phase from the absorbing one, belongs to directed percolation (DP) universality class. We claim that the conserved version with λ = 1 and σ = 0 also shows a DP transition; the apparent non-DP behaviour observed earlier is an artifact of undershooting in the decay of activity density starting from a random initial condition.

Radial Domany-Kinzel models with mutation and selection

We study the effect of spatial structure, genetic drift, mutation, and selective pressure on the evolutionary dynamics in a simplified model of asexual organisms colonizing a new territory. Under an appropriate coarse-graining, the evolutionary dynamics is related to the directed percolation processes that arise in voter models, the Domany-Kinzel (DK) model, contact process, and so on. We explore the differences between linear (flat front) expansions and the much less familiar radial (curved front) range expansions. For the radial expansion, we develop a generalized, off-lattice DK model that minimizes otherwise persistent lattice artifacts. With both simulations and analytical techniques, we study the survival probability of advantageous mutants, the spatial correlations between domains of neutral strains, and the dynamics of populations with deleterious mutations. "Inflation" at the frontier leads to striking differences between radial and linear expansions. For a colony with initial radius R(0) expanding at velocity v, significant genetic demixing, caused by local genetic drift, occurs only up to a finite time t(*)=R(0)/v, after which portions of the colony become causally disconnected due to the inflating perimeter of the expanding front. As a result, the effect of a selective advantage is amplified relative to genetic drift, increasing the survival probability of advantageous mutants. Inflation also modifies the underlying directed percolation transition, introducing novel scaling functions and modifications similar to a finite-size effect. Finally, we consider radial range expansions with deflating perimeters, as might arise from colonization initiated along the shores of an island.

Phase diagram and critical behavior of a forest-fire model in a gradient of immunity

The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl [Physica A 199, 183 (1993)], in which each site of a lattice can be in three possible states: occupied by a tree, empty, or occupied by a burning tree (fire). The trees grow at empty sites with probability p, healthy trees catch fire from adjacent burning trees with probability (1-g), where g is the immunity, and a burning tree becomes an empty site spontaneously. In this paper we study the FFMIT by means of the recently proposed gradient method (GM), considering the immunity as a uniform gradient along the horizontal axis of the lattice. The GM allows the simultaneous treatment of both the active and the inactive phases of the model in the same simulation. In this way, the study of a single-valued interface gives the critical point of the active-absorbing transition, whereas the study of a multivalued interface brings the percolation threshold into the active phase. Therefore we present a complete phase diagram for the FFMIT, for all range of p, where, besides the usual active-absorbing transition of the model, we locate a transition between the active percolating and the active nonpercolating phases. The average location and the width of both interfaces, as well as the absorbing and percolating cluster densities, obey a scaling behavior that is governed by the exponent α=1/(1+ν), where ν is the suitable correlation length exponent (ν(⊥) for the directed percolation transition and ν for the standard percolation transition). We also show that the GM allows us to calculate the critical exponents associated with both the order parameter of the absorbing transition and the number of particles in the multivalued interface. Besides, we show that by using the gradient method, the collapse in a single curve of cluster densities obtained for samples of different side is a very sensitive method in order to obtain the critical points and the percolation thresholds.

On the maximal noise for stochastic and QCD travelling waves

Using the relation of a set of nonlinear Langevin equations to reaction–diffusion processes, we note the existence of a maximal strength of the noise for the stochastic travelling wave solutions of these equations. Its determination is obtained using the field-theoretical analysis of branching-annihilation random walks near the directed percolation transition. We study its consequence for the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation. For the related Langevin equation modeling the quantum chromodynamic nonlinear evolution of gluon density with rapidity, the physical maximal-noise limit may appear before the directed percolation transition, due to a shift in the travelling-wave speed. In this regime, an exact solution is known from a coalescence process. Universality and other open problems and applications are discussed in the outlook.

Langevin Description of Critical Phenomena with Two Symmetric Absorbing States

On the basis of general considerations, we propose a Langevin equation accounting for critical phenomena occurring in the presence of two symmetric absorbing states. We study its phase diagram by mean-field arguments and direct numerical integration in physical dimensions. Our findings fully account for and clarify the intricate picture known so far from the aggregation of partial results obtained with microscopic models. We argue that the direct transition from disorder to one of two absorbing states is best described as a (generalized) voter critical point and show that it can be split into an Ising and a directed percolation transition in dimensions larger than one.

Directed polymer ? directed percolation transition: the strong disorder case

The transition of physical properties in disordered systems from strong disorder characteristics to weak disorder characteristics is studied for the directed polymer case. It is shown analytically that this transition is governed by the ratio ρ(pc)/k, where ρ is the probability density of the maximal bond of the optimal Min-Max path, pc is the critical probability of directed percolation, and k is the degree of disorder. This analytic result is found to be in agreement with numerical results related to this transition.

Role of diffusion in branching and annihilation random walk models

Different branching and annihilating random walk models are investigated by the cluster mean-field method and simulations in one and two dimensions. In the case of the A-->2A , 2A--> 0 model the cluster mean-field approximations show diffusion dependence in the phase diagram as was found recently by the nonperturbative renormalization group method [Phys. Rev. Lett. 92, 255703 (2004)]]. The same type of survey for the A-->2A , 4A--> 0 model results in a reentrant phase diagram, similar to that of the 2A-->3A , 4A--> 0 model [Phys. Rev. E 69, 036112 (2004)]]. Simulations of the A-->2A , 4A--> 0 model in one and two dimensions confirm the presence of both the directed percolation transitions at finite branching rates and the mean-field transition at zero branching rate. In two dimensions the directed percolation transition disappears for strong diffusion rates. These results disagree with the predictions of the perturbative renormalization group method.

Dynamic transition in etching with poisoning.

We study a lattice model for etching of a crystalline solid including the deposition of a poisoning species. The model considers normal and lateral erosion of the columns of the solid by a flux of etching particles and the blocking effects of impurities formed at the surface. As the probability p of formation of this poisoning species increases, the etching rate decreases and a continuous transition to a pinned phase is observed. The transition is in the directed percolation (DP) class, with the fraction of the exposed columns as the order parameter. This interpretation is consistent with a mapping of the interface problem in d+1 dimensions onto a d-dimensional contact process, and is confirmed by numerical results in d=1 and d=2. In the etching phase, the interface width scales with Kardar-Parisi-Zhang (KPZ) exponents, and shows a crossover from the critical DP behavior (W approximately t) to KPZ near the critical point, at etching times of the order of (pc-p)(-nu(||)). Anomalous roughening is observed at criticality, with the roughness exponent related to DP exponents as alphac=nu(||)/nu(perpendicular)>1. The main differences from previously studied DP transitions in growth models and isotropic percolation transitions in etching models are discussed. Investigations in real systems are suggested.

Depinning transitions in interface growth models

Pinning-depinning transitions are roughening transitions separating a growing phase and pinned (or blocked) one, and are frequently connected to transitions into absorbing states. In this review, we discuss lattice growth models exhibiting this type of dynamic transition. Driven growth in media with impurities, the competition between deposition and desorption and deposition of poisoning species are some of the physical mechanisms responsible for the transitions, leading to different types of stochastic growth rules. The growth models are classified according to the those mechanisms and possible applications are shown, which include suggestions of experimental realizations of directed percolation transitions.

One-dimensional nonequilibrium kinetic Ising models with local spin symmetry breaking: N-component branching annihilating random-walk transition at zero branching rate.

The effects of locally broken spin symmetry are investigated in one-dimensional nonequilibrium kinetic Ising systems via computer simulations and cluster-mean-field calculations. Besides a line of directed percolation transitions, a line of transitions belonging to N-component, two-offspring branching annihilating random-walk class (N-BARW2) is revealed in the phase diagram at zero branching rate. In this way a spin model for N-BARW2 transitions is proposed.

Fermi Partition Functions of Friendly Walkers and Pair Connectedness of Directed Percolation

Non-crossing random walkers with attractive interactions called friendly walkers (FWs) are studied. A restriction on trajectories, which is analogous to Pauli's exclusion principle, is imposed and the Fermi partition functions are defined. We prove a theorem that the pair connectedness of the bond directed percolation (DP) with bond concentration p is related to the Fermi grand partition function of FW if we set the temperature T =-1/( k B ln p ) and the chemical potential µ=- i π/ ln p , where k B is the Boltzmann constant and \({\rm i}=\sqrt{-1}\). The pure imaginary chemical potential means that the DP transition can be regarded as a symmetry breaking of parity in the number of FWs. As a corollary of the theorem, a new method is proposed for calculating the series expansion of the pair connectedness and percolation probability of DP using the low-temperature expansion data of FW.

Percolation transitions and wetting transitions in stochastic models

Stochastic models with irreversible elementary processes are introduced, and their macroscopic behaviors in the infinite-time and infinite-volume limits are studied extensively, in order to discuss nonequilibrium stationary states and phase transitions. The Domany-Kinzel model is a typical example of such an irreversible particle system. We first review this model, and explain that in a certain parameter region, the nonequilibrium phase transitions it exhibits can be identified with directed percolation transitions on the spatio-temporal plane. We then introduce an interacting particle system with particle conservation called friendly walkers (FW). It is shown that the m = 0 limit of the correlation function of m friendly walkers gives the correlation function of the Domany- Kinzel model, if we choose the parameters appropriately. We show that FW can be considered as a model of interfacial wetting transitions, and that the phase transitions and critical phenomena of FW can be studied using Fisher's theory of phase transitions in linear systems. The FW model may be the key to constructing a unified theory of directed percolation transitions and wetting transitions. Descriptions of FW as a model of interacting vicious walkers and as a vertex model are also given.