Taming Griffiths' singularities: Infinite differentiability of quenched correlation functions

Dreifus, Henrique von; Klein, Abel; Perez, J. Fernando

doi:10.1007/bf02099437

Cited by 47 publications

(52 citation statements)

References 12 publications

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“…Multi-scale methods have been employed at various occasions, such as for one-phase models in the presence of Griffiths singularities or for the random field Ising model. (9,10,12,18,21,28) In contrast to the usual case of cluster expansions one does not obtain analyticity (which may not even be valid). In our approach, we loosely follow the ideas of Fröhlich and Imbrie.…”

Section: Remark 42mentioning

confidence: 82%

On the Ising Model with Random Boundary Condition

Enter

Netočný²,

Schaap

2005

J Stat Phys

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We study the nearest-neighbour Ising model with a class of random boundary conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions 4 and higher that almost surely the only limit points for a sequence of increasing cubes are the plus and the minus state. For d=2 and d=3 we prove a similar result for sparse sequences of increasing cubes. This question was raised by Newman and Stein. Our results imply that the Newman-Stein metastate is concentrated on the plus and the minus state.

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Section: Remark 42mentioning

confidence: 82%

On the Ising Model with Random Boundary Condition

Enter

Netočný²,

Schaap

2005

J Stat Phys

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“…Note that a similar construction was used in deriving (2.16) and (2.17) in [16], see also lemma 3.1 in [13].…”

Section: -7mentioning

confidence: 99%

On thermodynamic states of the Ising model on scale-free graphs

Kozitsky¹

2013

Condens. Matter Phys.

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There is proposed a model of scale-free random graphs which are locally close to the uncorrelated complex random networks with divergent 〈k 2 〉 studied in, e.g., Dorogovtsev S.N. et al., Rev. Mod. Phys., 2008, 80, 1275 It is shown that the Ising model on the proposed graphs with interaction intensities of arbitrary signs with probability one is in a paramagnetic state at sufficiently high finite values of the temperature. For the same graphs, the bond percolation model with probability one is in a nonpercolative state for positive values of the percolation probability. These results and their possible extensions are also discussed.

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“…However if the probability that the loglikelihood takes unbounded values is very small, so that the domains with large loglikelihood do not percolate, one expects that the correlation still decays on average (over the noise realizations and code ensemble). There are a number of methods to address the issue of uniqueness of Gibbs measure and correlation decay in random spin systems when there are unbounded interactions [10], [11], [12]. Here we use an expansion technique that goes back to Dreifus, Klein and Perez [12] to show the correlation decay for general BMS channels in the high noise regime.…”

Section: B Correlation Decaymentioning

confidence: 99%

“…There are a number of methods to address the issue of uniqueness of Gibbs measure and correlation decay in random spin systems when there are unbounded interactions [10], [11], [12]. Here we use an expansion technique that goes back to Dreifus, Klein and Perez [12] to show the correlation decay for general BMS channels in the high noise regime. In general these expansions are non trivial because, although the domains with large loglikelihoods do not percolate their size is unbounded, so there are arbitrarily large regions of the graph where one does not expand around the "correct point".…”

Section: B Correlation Decaymentioning

confidence: 99%

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Proof of replica formulas in the high noise regime for communication using LDGM codes

Kudekar

Macris

2008

2008 IEEE Information Theory Workshop

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Abstract-We consider communication over a binary input memoryless output symmetric channel with low density generator matrix codes and optimal maximum a posteriori decoding. It is known that the problem of computing the average conditional entropy, over such code ensembles in the asymptotic limit of large block length, is closely related to computing the free energy of a mean field spin glass in the thermodynamic limit. Tentative explicit formulas for these quantities have been derived thanks to the replica method (of spin glass theory) and are generally conjectured to be exact. In this contribution we show that the replica solution is indeed exact in the high noise regime, where it coincides with density evolution equations. Our method uses ideas coming from high temperature expansions in spin glass theory. I. MOTIVATIONWe consider communication over a binary input memoryless output symmetric channel (BMS) with low density generator matrix (LDGM) codes and optimal maximum a posteriori (MAP) decoding. Let U 1 , U 2 , . . . , U n denote the information bits from which we create m generator bits X 1 , X 2 , . . . , X m using an LDGM code. The generator bits are transmitted through the channel with transition probability p Y |X and Y 1 , Y 2 , . . . , Y m is received. We are interested in computing the average (over the code ensemble) entropy of the information word U n given the received word Y m . The average over the code ensemble conditional entropyof the transmitted information word U n conditional to the received message Y m can be formally computed by the ill-defined replica method of statistical mechanics. From the rather explicit formula obtained in this way one may also compute its derivative with respect to the noise level, a quantity that is also called MAP GEXIT curve [1]. It is believed that replica symmetry breaking is absent for symmetric channels (for bit MAP decoding) and is conjectured that the replica symmetric equations are rigorously exact. It is well known that away from the intervals between BP and MAP thresholds, the replica formulas coincide with the ones given by density evolution. Thus the conjecture also tells that density evolution gives the exact conditional entropy and MAP GEXIT curve away from intervals separating BP and MAP thresholds While the general proof of this conjecture is still an open problem, some progress has been made in the last years. Tight bounds have been derived using a variety of tools (physical In this paper we provide a full proof of the conjecture in high noise regimes for the case of general LDGM code ensembles with bounded degrees and BIAWGN, BEC, BSC channels (and convex combinations of them). We believe that our proof can be extended to a more general class of BMS channels although some of the estimates become more technical. Apart from this result the interest of this work also lies in the method which departs from all the ones previously used. Our main tool is an expansion, that has its roots in high temperature expansions of statistical mechanics, and al...

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Taming Griffiths' singularities: Infinite differentiability of quenched correlation functions

Cited by 47 publications

References 12 publications

On the Ising Model with Random Boundary Condition

On the Ising Model with Random Boundary Condition

On thermodynamic states of the Ising model on scale-free graphs

Proof of replica formulas in the high noise regime for communication using LDGM codes

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