Correlations and critical behavior of the q-model

TL;DR

This paper derives exact correlation functions for the q-model, revealing universal scaling behavior near a critical point, with critical exponents analogous to thermodynamic phase transitions.

Contribution

It provides the first exact expressions for correlation functions in the 2D q-model and characterizes its critical scaling behavior.

Findings

Correlation functions have a universal scaling form near criticality

Critical exponents $ u$ and $z$ are determined

Scaling function and critical behavior are analogous to thermodynamic phenomena

Abstract

The q-model is a random walk model used to describe the flow of stress in a stationary granular medium. Here we derive the exact horizontal and vertical correlation functions for the q-model in two dimensions. We show that close to a critical point identified in earlier work these correlation functions have a universal scaling form reminiscent of thermodynamic critical phenomena. We determine the form of the universal scaling function and the associated critical exponents $\nu$ and $z$.