Directed percolation: a finite-size renormalisation group approach

Supervised and unsupervised learning of directed percolation

Machine learning (ML) has been well applied to studying equilibrium phase transition models by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into nonequilibrium phase transitions. The extra dimension in a given nonequilibrium system, namely time, can greatly slow down the procedure toward the steady state. In this paper we find that by using some simple techniques of ML, non-steady-state configurations of directed percolation (DP) suffice to capture its essential critical behaviors in both (1+1) and (2+1) dimensions. With the supervised learning method, the framework of our binary classification neural networks can identify the phase transition threshold, as well as the spatial and temporal correlation exponents. The characteristic time t_{c}, specifying the transition from active phases to absorbing ones, is also a major product of the learning. Moreover, we employ the convolutional autoencoder, an unsupervised learning technique, to extract dimensionality reduction representations and cluster configurations of (1+1) bond DP. It is quite appealing that such a method can yield a reasonable estimation of the critical point.

Onset of meso-scale turbulence in active nematics

Meso-scale turbulence is an innate phenomenon, distinct from inertial turbulence, that spontaneously occurs at low Reynolds number in fluidized biological systems. This spatiotemporal disordered flow radically changes nutrient and molecular transport in living fluids and can strongly affect the collective behaviour in prominent biological processes, including biofilm formation, morphogenesis and cancer invasion. Despite its crucial role in such physiological processes, understanding meso-scale turbulence and any relation to classical inertial turbulence remains obscure. Here we show how the motion of active matter along a micro-channel transitions to meso-scale turbulence through the evolution of locally disordered patches (active puffs) from an ordered vortex-lattice flow state. We demonstrate that the stationary critical exponents of this transition to meso-scale turbulence in a channel coincide with the directed percolation universality class. This finding bridges our understanding of the onset of low-Reynolds-number meso-scale turbulence and traditional scale-invariant turbulence in confinement.

Simulating the contact process in heterogeneous environments

The one-dimensional contact process (CP) in a heterogeneous environment-a binary chain consisting of two types of site with different recovery rates-is investigated. It is argued that the commonly used random-sequential Monte Carlo simulation method which employs a discrete notion of time is not faithful to the rates of the contact process in a heterogeneous environment. Therefore, a modification of this algorithm along with two alternative continuous-time implementations are analyzed. The latter two are an adapted version of the n -fold way used in Ising model simulations and a method based on a modified priority queue. It is demonstrated that the commonly used (but incorrect as we believe) discrete-time method yields a different critical threshold from all other algorithms considered. Finite-size scaling of the lowest gap in the spectrum of the Liouville time-evolution operator for the CP gives an estimate of the critical rate which supports these findings. Further, a performance test indicates an advantage in using the continuous-time methods in systems with heterogeneous rates. This result promises to help in the analysis of the CP in disordered systems with heterogeneous rates in which simulation is a challenging task due to very long relaxation times.

Active width at a slanted active boundary in directed percolation

The width W of the active region around an active moving wall in a directed percolation process diverges at the percolation threshold p(c) as W approximately Aepsilon(-nu( parallel)) ln(epsilon(0)/epsilon), with epsilon=p(c)-p, epsilon(0) a constant, and nu( parallel)=1.734 the critical exponent of the characteristic time needed to reach the stationary state xi( parallel) approximately epsilon(-nu(parallel)). The logarithmic factor arises from screening the statistically independent needle shaped subclusters in the active region. Numerical data confirm this scaling behavior.

Upper bounds for the critical probability of oriented percolation in two dimensions

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.