Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDirected percolation and directed animals
175
0
0.0
(
0
)
Authors
Deepak Dhar
Ask ChatGPT about the research
No Arabic abstract
These lectures provide an introduction to the directed percolation and directed animals problems, from a physicists point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation to undirected first passage percolation problem are described. Equivalence of the $d$-dimensional directed animals problem to $(d-1)$-dimensional Yang-Lee edge-singularity problem is established. Self-organized critical formulation of the percolation problem, which does not involve any fine-tuning of coupling constants to get critical behavior is briefly discussed.
rate research
Read More
We study a hierarchy of directed percolation (DP) processes for particle species A, B, ..., unidirectionally coupled via the reactions A -> B, ... 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 >= 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 <= 3.
We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the augmented triangular lattice shows a continuous transition at a non-trivial p_c. On the augmented triangular lattice we find, by extensive numerical simulation, that the directed rigidity percolation transition belongs to the same universality class as directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e. the spreading of rigidity from finite clusters close to a non-rigid wall. Near the discontinuous transition at p=1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites analytically. Our results are confirmed by numerical simulation.
Conserved directed-percolation (C-DP) and the depinning transition of a disordered elastic interface belong to the same universality class as has been proven very recently by Le Doussal and Wiese [Phys. Rev. Lett.~textbf{114}, 110601 (2015)] through a mapping of the field theory for C-DP onto that of the quenched Edwards-Wilkinson model. Here, we present an alternative derivation of the C-DP field theoretic functional, starting with the coherent state path integral formulation of the C-DP and then applying the Grassberger-transformation, that avoids the disadvantages of the so-called Doi-shift. We revisit the aforementioned mapping with focus on a specific term in the field theoretic functional that has been problematic in the past when it came to assessing its relevance. We show that this term is redundant in the sense of the renormalization group.
We study critical spreading in a surface-modified directed percolation model in which the left- and right-most sites have different occupation probabilities than in the bulk. As we vary the probability for growth at an edge, the critical exponents switch from the compact directed percolation class to ordinary directed percolation. We conclude that the nonuniversality observed in models with multiple absorbing configurations cannot be explained as a simple surface effect.
A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behaviour observed along the transition line changes from a directed-percolation to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions, Mean-field arguments and the mapping on a yet simpler model provide some further insight on the overall scenario.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
