Scaling Relations For The CLG's Critical Exponents
Clément Erignoux, Assaf Shapira, Marielle Simon
TL;DR
This work analyzes the constrained lattice gas (CLG) and its self-organized critical behavior across dimensions, focusing on how macroscopic observables scale near the critical density $\rho_c$ and how critical exponents relate through scaling relations. The authors establish a framework linking active density $\rho_a$, diffusion $D$, correlation lengths $\xi_\times$ and $\xi_\perp$, compressibility $\chi$, and conductivity $\sigma$, deriving identities such as $\alpha = \beta - 1$ and $\gamma = \nu_\times (d - 2\zeta)$, along with an Einstein-type relation $\sigma = D\chi$. They extend the analysis to boundary-driven CLG to obtain explicit stationary current expressions in finite domains, and in $d=1$ provide exact grand-canonical states and universal exponents $b=\beta=\gamma=\nu_\times=\nu_\perp=1$ with $\alpha=0$, plus explicit forms for $\rho_a(\rho)$ and other observables. The results offer a rigorous-leaning yet practically useful map from microscopic constraints to macroscopic scaling laws, shedding light on the nature of criticality and transport in constrained lattice gases.
Abstract
We consider, in any dimension, the constrained lattice gas introduced by Rossi et al., which is an exclusion process on a d-dimensional lattice following the additional constraint that only particles with at least one occupied neighbour can jump. In dimension d > 2, this model features self-organized criticality at some critical density of particles. Numerical simulations predict the existence of scaling exponents close to criticality, and several relations can be derived between these exponents. The goal of this article is to give a mathematical framework for these relations, which have been numerically established in a companion article.