Extension of Thomas’ result and upper bound on the spectral gap of $d(\ge 3)$-dimensional Stochastic Ising models

Sugimine, Nobuaki

doi:10.1215/kjm/1250284715

Cited by 5 publications

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“…Theorem 4.2(a) was obtained in a weaker form by Higuchi and Yoshida in [46] and then in the present form by Alexander and Yoshida [47]. Theorem 4.2(b) is due to Sugimine [48].…”

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confidence: 70%

Phase Transition From the Viewpoint of Relaxation Phenomena

Yoshida

2003

Rev. Math. Phys.

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“…Theorem 4.2(a) was obtained in a weaker form by Higuchi and Yoshida in [46] and then in the present form by Alexander and Yoshida [47]. Theorem 4.2(b) is due to Sugimine [48].…”

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confidence: 70%

Phase Transition From the Viewpoint of Relaxation Phenomena

Yoshida

2003

Rev. Math. Phys.

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“…A good instance of the latter situation is represented by the Glauber dynamics for the usual ±1 Ising model at low temperature in the absence of an external magnetic field (see Section 1.2). When the system is analyzed in a finite box of side L of the d-dimensional lattice Z d with free boundary conditions, the relaxation to the Gibbs reversible measure occurs on a time scale exponentially large in the surface L d−1 [27,26] because of the energy barrier between the two stable phases of the system (see Section 1.3 for a more quantitative statement). When instead one of the two phases is selected by homogeneous boundary conditions, e.g.…”

Section: Introduction Model and Main Resultsmentioning

confidence: 99%

On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

Martinelli

Toninelli

2009

Commun. Math. Phys.

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We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to −). For β large enough we show that for any ε > 0 there exists c = c(β, ε) such that the corresponding mixing time Tmix satisfies limL→∞ P (Tmix ≥ exp(cL ε )) = 0. In the non-random case τ ≡ + (or τ ≡ −), this implies that Tmix ≤ exp(cL ε ). The same bound holds when the boundary conditions are all + on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior Tmix = O(L 2 ), considerably improves upon the previous known estimates of the form Tmix ≤ exp(cL 1 2 +ε ). The techniques are based on induction over length scales, combined with a judicious use of the so-called "censoring inequality" of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.2000 Mathematics Subject Classification: 60K35, 82C20 and the right-hand side is an increasing function of σ; in accord with the notations of Section 1.2, L + J denotes the generator of the dynamics in J with + boundary conditions on ∂J (its invariant measure is of course π + J ) and L is the generator of the infinite-volume dynamics. One has then (using once more monotonicity)f 2 (4.23) which, together with (4.21), gives ρ(t) = Var + ∞ e tL f ≤ Var π + J e tL + J f + c e −c . (4.24) By (1.6), one has that Var π + J e tL + J f ≤ Var π + J

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“…However, even their effect is limited as time progresses. It is known that in a finite box of side length L of the d-dimensional lattice Z d with free boundary conditions, the relaxation to the Gibbs measure occurs on a time scale exponentially large in the surface L d−1 [25], [26]. Thus, for the 2D grid, the chain mixes after e Θ( √ n) time, at which point all information stored in the system is lost.…”

Section: Stripe-based Achievability Schemementioning

confidence: 99%

Information Storage in the Stochastic Ising Model at Low Temperature

Goldfeld

Bresler

Polyanskiy

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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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.

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Extension of Thomas’ result and upper bound on the spectral gap of $d(\ge 3)$-dimensional Stochastic Ising models

Cited by 5 publications

References 0 publications

Phase Transition From the Viewpoint of Relaxation Phenomena

Phase Transition From the Viewpoint of Relaxation Phenomena

On the Mixing Time of the 2D Stochastic Ising Model with “Plus” Boundary Conditions at Low Temperature

Information Storage in the Stochastic Ising Model at Low Temperature

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