Information Storage in the Stochastic Ising Model at Low Temperature

Motivated by questions of data stabilization in emerging magnetic storage technologies, we study the retention of information in interacting particle systems. The interactions between particles adhere to the stochastic Ising model (SIM) on the two-dimensional (2D) \sqrt n × \sqrt n grid. The measure of interest is the information capacity {I-n}\left(t\right) \triangleq {\max -{{p-{{X-0}}}}}I\left({{X-0};{X-t}}\right), where the initial spin configuration X₀ is a user-controlled input and the output configuration X_t is produced by running t steps of Glauber dynamics. After the results on the zero-temperature regime reported last year, this work focuses on the positive but low temperature regime. We first show that storing more than a single bit for an exponential time is impossible when the initial configuration is drawn from the equilibrium distribution. Specifically, if X₀ is drawn according to the Gibbs measure, then I(X₀; X_t) ≤ 1 + o(1) for t \geq \exp \left({c{n^{\frac{1}{4} + \varepsilon }}}\right). On the other hand, when scaling time with β, we propose a stripe-based coding scheme that stores order of \sqrt n bits for exp(β) time. Key to the analysis of the scheme is a new result on the survival time of a single plus-labeled stripe in a sea of minuses. Together, the 1-bit upper bound and the striped-based storage scheme constitute initial steps towards a general analysis of I_n(t) for β > 0.