Exact first-passage exponents of 1D domain growth: relation to a reaction diffusion model

TL;DR

This paper derives the exact first-passage exponents for one-dimensional domain growth in zero-temperature Glauber dynamics of the Ising and Potts models, revealing their relation to a reaction-diffusion process.

Contribution

It provides the exact calculation of the decay exponent of unflipped spins in 1D, connecting domain growth to an exactly solvable coagulation model.

Findings

Exact expression for the exponent (q) in 1D

Demonstrates the relation between domain growth and coagulation models

Provides insights into spin flip dynamics in zero-temperature Ising and Potts models

Abstract

In the zero temperature Glauber dynamics of the ferromagnetic Ising or $q$-state Potts model, the size of domains is known to grow like $t^{1/2}$. Recent simulations have shown that the fraction $r(q,t)$ of spins which have never flipped up to time $t$ decays like a power law $r(q,t) \sim t^{-\theta(q)}$ with a non-trivial dependence of the exponent $\theta(q)$ on $q$ and on space dimension. By mapping the problem on an exactly soluble one-species coagulation model ($A+A\rightarrow A$), we obtain the exact expression of $\theta(q)$ in dimension one.