Most information systems store data by modifying the local state of matter, in the hope that atomic (or sub-atomic) 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 $X_0$ 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 of the system $n$. For the zero-temperature SIM on the two-dimensional $\sqrt{n}\times\sqrt{n}$ grid and free boundary conditions, it is easy to show that $I_n(t) = \Theta(n)$ for $t=O(n)$. In addition, we show that on the order of $\sqrt{n}$ bits can be stored for infinite time in striped configurations. The $\sqrt{n}$ achievability is optimal when $t\to\infty$ and $n$ is fixed. One of the main results of this work is an achievability scheme that stores more than $\sqrt{n}$ bits (in orders of magnitude) for superlinear (in $n$) times. 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. 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 $t\geq e^{cn^{\frac{1}{4}+\epsilon}}$. On the other hand, when scaling time with $\beta$, 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 $\beta$.

中文翻译：

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随机 Ising 模型中的信息存储

大多数信息系统通过修改物质的局部状态来存储数据，希望原子（或亚原子）局部相互作用能够在足够长的时间内稳定状态，从而允许以后恢复。在这项工作中，我们开始研究本地交互系统中的信息保留。相互作用粒子的时间演化通过随机 Ising 模型 (SIM) 进行建模。初始旋转配置 $X_0$ 用作用户控制的输入。输出配置 $X_t$ 是通过运行 Glauber 链的 $t$ 步骤产生的。我们的主要目标是评估信息容量 $I_n(t)\triangleq\max_{p_{X_0}}I(X_0;X_t)$ 当时间 $t$ 与系统 $n$ 的大小成比例时。对于二维 $\sqrt{n}\times\sqrt{n}$ 网格上的零温度 SIM 和自由边界条件，很容易证明 $I_n(t) = \Theta(n)$ 对于 $t=O(n)$。此外，我们展示了 $\sqrt{n}$ 的顺序可以在条带配置中无限存储。当 $t\to\infty$ 和 $n$ 固定时，$\sqrt{n}$ 的可实现性是最佳的。这项工作的主要结果之一是可实现性方案，该方案可存储超过 $\sqrt{n}$ 位（数量级）超线性（$n$）次。该方案的分析将系统分解为 $\Omega(\sqrt{n})$ 独立的 Z 通道，其交叉概率是通过（最近严格建立的）Lifshitz 相边界运动定律找到的。我们还提供了积极但小的温度范围的结果。我们表明，根据 Gibbs 度量绘制的初始配置不能为 $t\geq e^{cn^{\frac{1}{4}+\epsilon}}$ 保留多于一个位。