Information Storage in the Stochastic Ising Model at Low Temperature

Abstract

Most information systems store data by modifying the local state of matter, in the hope that atomic (or subatomic) local interactions would stabilize the state for a sufficiently long time, thereby allowing later recovery. In this work we initiate the study of information retention in locally-interacting systems. The evolution in time of the interacting particles is modeled via the stochastic Ising model (SIM). The initial spin configuration X0 serves as the user-controlled input. The output configuration Xt is produced by running t steps of the Glauber chain. Our main goal is to evaluate the information capacity In(t) maxp X 0 I(X0; Xt) when the time t scales with the size of the system n. For the zero-temperature SIM on the two-dimensional √ n × √ n grid and free boundary conditions, it is easy to show that In(t) = Θ(n) for t = O(n). In addition, we show that on the order of √ n bits can be stored for infinite time (and even with zero error) in horizontally or vertically striped configurations. The √ n achievability is optimal when t → ∞ and n is fixed.One of the main results of this work is an achievability scheme that stores more than √ n bits (in orders of magnitude) for superlinear (in n) times. The analysis of the scheme decomposes the system into Ω( √ n) independent Zchannels whose crossover probability is found via the (recently rigorously established) Lifshitz law of phase boundary movement. Lastly for the zero-temperature case, two order optimal characterizations of In(t), for all t, are given for the grid dynamics with an external magnetic field and for the dynamics over the honeycomb lattice. In both these setups In(t) = Θ(n), for all t, suggesting their superiority over the grid without an external field for storage purposes. We also provide results for the positive but small temperature regime. We show that an initial configuration drawn according to the Gibbs measure cannot retain more than a single bit for t ≥ e cn 1 4 +ǫ . On the other hand, when scaling time with β, the stripe-based coding scheme (that stores for infinite time at zero temperature) is shown to retain its bits for time that is exponential in β.