Replica Method and Finite Volume Corrections

Campellone, Matteo; Parisi, G.; Virasoro, M. A.

doi:10.1007/s10955-009-9891-1

Cited by 18 publications

(28 citation statements)

References 22 publications

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“…It would be interesting to look at models which do not have built in ultrametricity to see at what level of the finite size corrections the non-ultrametric structure becomes apparent [28,29,30,31]. More challenging would be to understand the generality or the limitations of corrections such as (5,6) in the broader context of mean field models of disordered systems.…”

Section: Discussionmentioning

confidence: 99%

Finite Size Corrections to the Parisi Overlap Function in the GREM

Derrida

Mottishaw

2018

J Stat Phys

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We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model (GREM) in two situations where replica symmetry is broken in the thermodynamic limit. Our calculations do not use replicas, but shed some light on what the replica method should give for finite size corrections. In the gradual freezing situation, which is known to exhibit full replica symmetry breaking, we show that the finite size corrections lead to a modification of the simple relations between the sample averages of the overlaps Y k between k configurations predicted by replica theory. This can be interpreted as fluctuations in the replica block size with a negative variance. The mechanism is similar to the one we found recently in the random energy model [1]. We also consider a simultaneous freezing situation, which is known to exhibit one step replica symmetry breaking. We show that finite size corrections lead to full replica symmetry breaking and give a more complete derivation of the results presented in [2] for the directed polymer on a tree.

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Section: Discussionmentioning

confidence: 99%

Finite Size Corrections to the Parisi Overlap Function in the GREM

Derrida

Mottishaw

2018

J Stat Phys

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“…Formulas (22)(23)(24)(25) are exact for the Poisson REM. They summarize all the previous ones in particular (17,20).…”

Section: The Moments Of Z and The Weighted Overlapsmentioning

confidence: 99%

Finite size corrections in the random energy model and the replica approach

Derrida¹,

Mottishaw²

2015

J. Stat. Mech.

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We present a systematict and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as "complex replica numbers". Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the "complex replica numbers" near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

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“…It has been suggested [57,58,59] that one then must consider fluctuations from the saddle point, in particular, the size of clusters m explicitly.…”

Section: Appendix G: Fluctuations Of Clusters In One-step Rsb Solutionsmentioning

confidence: 99%

“…In particular in the case of REM the overlap between the states can be either 0 or 1 by the construction of the model so that fluctuation of the size of clusters amount to very large fluctuations of the amplitudes of the elements of the Parisi matrix [59].…”

Section: Appendix G: Fluctuations Of Clusters In One-step Rsb Solutionsmentioning

confidence: 99%

Stepwise responses in mesoscopic glassy systems: A mean-field approach

Yoshino

Rizzo

2008

Phys. Rev. B

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We study statistical properties of peculiar responses in glassy systems at mesoscopic scales based on a class of mean-field spin-glass models which exhibit 1 step replica symmetry breaking. Under variation of a generic external field, a finite-sized sample of such a system exhibits a series of step wise responses which can be regarded as a finger print of the sample. We study in detail the statistical properties of the step structures based on a low temperature expansion approach and a replica approach. The spacings between the steps vanish in the thermodynamic limit so that arbitrary small but finite variation of the field induce infinitely many level crossings in the thermodynamic limit leading to a static chaos effect which yeilds a self-averaging, smooth macroscopic response. We also note that there is a strong analogy between the problem of step-wise responses in glassy systems at mesoscopic scales and intermittency in turbulent flows due to shocks.

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Replica Method and Finite Volume Corrections

Cited by 18 publications

References 22 publications

Finite Size Corrections to the Parisi Overlap Function in the GREM

Finite Size Corrections to the Parisi Overlap Function in the GREM

Finite size corrections in the random energy model and the replica approach

Stepwise responses in mesoscopic glassy systems: A mean-field approach

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