Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNon-fixation for Biased Activated Random Walks
306
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.
rate research
Read More
Lecture Notes. Minicourse given at the workshop Activated Random Walks, DLA, and related topics at IMeRA-Marseille, March 2015.
We consider Activated Random Walks on $Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles through the origin rescales to a pure-jump self-similar process which we describe explicitly.
Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
We consider symmetric activated random walks on $mathbb{Z}$, and show that the critical density $zeta_c$ satisfies $csqrt{lambda} leq zeta_c(lambda) leq C sqrt{lambda}$ where $lambda$ denotes the sleep rate.
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process `{it space-time Mittag-Leffler process}. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a `well-scaled diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the `state density kernel solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
