Information Storage in the Stochastic Ising Model at Low Temperature

Abstract

Most information systems store data by modifying the local state of the matter, in the hope that atomic (or subatomic) local interactions would stabilize the state for sufficiently long time, thereby allowing later recovery. In this work we initiate the study of information retention properties of locally-interacting systems. We model the time-dependent interactions between the different particles via the stochastic Ising model (SIM). The initial spin configuration $X_{0}$ serves as the user-controlled input. The output configuration $X_{t}$ is produced by running $t$ steps of the Glauber chain. Our main goal is to evaluate the information capacity $I_{n}(t) \triangleq\max_{p X_{0}}I({X}_{0}; X_{t})$ when the time $t$ scales with the size or the system $n$ according to various rates. For the zero-temperature SIM on the two-dimensional $\sqrt{n}\times\sqrt{n}$ grid and free boundary condition, it is easy to show that $I_{n}(t)=\Theta(n)$ as long as $t=O(n)$ . In addition, we show that order of $\sqrt{n}$ bits can be stored for infinite time (and even with zero error). The $\sqrt{n}$ achievability is optimal when $t\rightarrow\infty$ and $n$ is fixed. Our main result is in extending achievability to super-linear (in n) times via a coding scheme that reliably stores more than $\sqrt{n}$ bits (in orders of magnitude). The analysis of the scheme decomposes the system into $\Omega(\sqrt{n})$ independent Z-channels whose crossover probability is found via the (recently rigorously established) Lifshitz law of phase boundary movement. Finally, two order optimal characterizations of $I_{n}(t)$ , for all $t$ , are given for the grid dynamics with an external magnetic field and for the dynamics on the Honeycomb lattice. It shown that $I_{n}(t)=\Theta(n)$ in both cases, suggesting their superiority over the grid without an external field for storage purposes.