TY - JOUR

T1 - Information Storage in the Stochastic Ising Model

AU - Goldfeld, Ziv

AU - Bresler, Guy

AU - Polyanskiy, Yury

N1 - Funding Information:
Manuscript received May 8, 2018; revised June 20, 2019; accepted November 14, 2020. Date of publication January 4, 2021; date of current version February 17, 2021. The work of Ziv Goldfeld was supported in part by the National Science Foundation under Grant CCF-1947801, in part by the 2020 IBM Academic Award, and in part by the Rothschild Postdoctoral Fellowship. The work of Guy Bresler was supported in part by the Office of Naval Research (ONR) under Grant N00014-17-1-2147, in part by the Defense Advanced Research Projects Agency (DARPA) under Grant W911NF-16-1-0551, and in part by the NSF under Grant CCF-1565516. The work of Yury Polyanskiy was supported in part by the National Science Foundation CAREER Award under Grant CCF-12-53205, in part by the Center for Science of Information (CSoI), in part by the NSF Science and Technology Center under Grant CCF-09-39370, and in part by a grant from the Skoltech–MIT Joint Next Generation Program (NGP). This article was presented in part at the 2018 IEEE International Symposium on Information Theory (ISIT-2018), in part at the 2019 IEEE International Symposium on Information Theory (ISIT-2019), and in part at the 2020 Beyond IID Conference (BIID-2020). (Corresponding author: Ziv Goldfeld.) Ziv Goldfeld is with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14850 USA (e-mail: goldfeld@cornell.edu).
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - Most information storage devices write data by modifying the local state of matter, in the hope that sub-atomic local interactions stabilize the state for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution of physical states in magnetic storage media affects their capacity, this work initiates the study of information retention in locally-interacting particle systems. The system dynamics follow the stochastic Ising model (SIM) over a 2-dimensional sqrt {{n}}times sqrt {{n}} grid. The initial spin configuration {X}{0} serves as the user-controlled input. The output configuration {X}{{t}} is produced by running t steps of Glauber dynamics. Our main goal is to evaluate the information capacity {I}{{n}}({t}):=max {{p}{{X}{0}}}{I}({X}{0};{X}{{t}}) when time t scales with the system's size n. While the positive (but low) temperature regime is our main interest, we start by exploring the simpler zero-temperature dynamics. We first show that at zero temperature, order of sqrt {{n}} bits can be stored in the system indefinitely by coding over stable, striped configurations. While sqrt {{n}} is order optimal for infinite time, backing off to {t}< infty , higher orders of {I}{{n}}({t}) are achievable. First, via linear coding arguments imply we show that {I}{{n}}({t}) = Theta ({n}) for {t}={O}({n}). To go beyond the linear scale, we develop a droplet-based achievability scheme that reliably stores Omega left ({{n}/log {n}}right) for {t}={O}({n}log {n}) time ( log {n} can be replaced with any {o}({n}) function). Moving to the positive but low temperature regime, two main results are provided. First, we show that an initial configuration drawn from the Gibbs measure cannot retain more than a single bit for {t}geq exp ({C}beta {n}{1/4+epsilon }) time. On the other hand, when scaling time with the inverse temperature beta , the stripe-based coding scheme (that stores for infinite time at zero temperature) is shown to retain its bits for {e}{{c}beta }.

AB - Most information storage devices write data by modifying the local state of matter, in the hope that sub-atomic local interactions stabilize the state for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution of physical states in magnetic storage media affects their capacity, this work initiates the study of information retention in locally-interacting particle systems. The system dynamics follow the stochastic Ising model (SIM) over a 2-dimensional sqrt {{n}}times sqrt {{n}} grid. The initial spin configuration {X}{0} serves as the user-controlled input. The output configuration {X}{{t}} is produced by running t steps of Glauber dynamics. Our main goal is to evaluate the information capacity {I}{{n}}({t}):=max {{p}{{X}{0}}}{I}({X}{0};{X}{{t}}) when time t scales with the system's size n. While the positive (but low) temperature regime is our main interest, we start by exploring the simpler zero-temperature dynamics. We first show that at zero temperature, order of sqrt {{n}} bits can be stored in the system indefinitely by coding over stable, striped configurations. While sqrt {{n}} is order optimal for infinite time, backing off to {t}< infty , higher orders of {I}{{n}}({t}) are achievable. First, via linear coding arguments imply we show that {I}{{n}}({t}) = Theta ({n}) for {t}={O}({n}). To go beyond the linear scale, we develop a droplet-based achievability scheme that reliably stores Omega left ({{n}/log {n}}right) for {t}={O}({n}log {n}) time ( log {n} can be replaced with any {o}({n}) function). Moving to the positive but low temperature regime, two main results are provided. First, we show that an initial configuration drawn from the Gibbs measure cannot retain more than a single bit for {t}geq exp ({C}beta {n}{1/4+epsilon }) time. On the other hand, when scaling time with the inverse temperature beta , the stripe-based coding scheme (that stores for infinite time at zero temperature) is shown to retain its bits for {e}{{c}beta }.

KW - Glauber dynamics

KW - Markov chains

KW - information capacity

KW - stochastic ising model

KW - storage

UR - http://www.scopus.com/inward/record.url?scp=85099210735&partnerID=8YFLogxK

U2 - 10.1109/TIT.2020.3049028

DO - 10.1109/TIT.2020.3049028

M3 - Article

AN - SCOPUS:85099210735

VL - 67

SP - 1373

EP - 1399

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 3

M1 - 9312622

ER -