To each directed acyclic graph (this includes some D-dimensional lattices) one can associate some abelian algebras that we call directed abelian algebras (DAA). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground state wavefunctions (stationary states probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and choose Hamiltonians linear in the generators, in the finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent $z = D$. One possible application of the DAA is to sandpile models. In the paper we present this application considering one and two dimensional lattices. In the one dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent $σ_τ = 3/2$). We study the local densityof particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found $σ_τ = 1.782 \pm 0.005$.
Guangdong Technion-Israel Institute of Technology（广东以色列理工学院）；Technion-Israel Institute of Technology（以色列理工学院）；Indian Institute of Science Education and Research, Berhampur（印度科学教育研究所伯拉姆普尔分校）
Directed update for the Stochastic Green Function algorithm
在最近的出版物（Phys. Rev E 77，056705（2008））中，我们已经提出了随机格林函数（SGF）算法，其具有通用且易于应用于形式H=V-T的任何晶格哈密顿量的性质，其中V在所选择的占据数基中是对角的，并且T仅具有正矩阵元素。我们在这里提出了一个修改版本的更新计划，保持了简单性和通用性的原始SGF算法，并显着提高其效率。
We study the problem of identifying macroscopic structures in networks, characterizing the impact of introducing link directions on the detectability phase transition. To this end, building on the stochastic block model, we construct a class of hardly detectable directed networks. We find closed form solutions by using belief propagation method showing how the transition line depends on the assortativity and the asymmetry of the network. Finally, we numerically identify the existence of a hard phase for detection close to the transition point.
We introduce a natural stochastic extension, called SSP, of the abelian sandpile model(ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by the authors [5]. The importance of understanding how LLL works cannot be stresse…
Nonreversible Markov chains can outperform reversible chains in the Markov chain Monte Carlo method. Lifting is a versatile approach to introducing net stochastic flow in state space and constructing a nonreversible Markov chain. We present here an application of the lifting technique to the directed-worm algorithm. The transition probability of the worm update is optimized using the geometric allocation approach; the worm backscattering probability is minimized, and the stochastic flow breaking the detailed balance is maximized. We demonstrate the performance improvement over the previous wor…
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/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 tr…
