Some non-perturbative calculations on spin glasses

Campellone, Matteo

doi:10.1088/0305-4470/28/8/009

Cited by 7 publications

(10 citation statements)

References 10 publications

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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%

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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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“…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%

“…in the range where the contour C 1 crosses the real axis). If not one can deform the contour to pass through the saddle point but one should not forget the contribution of the poles at the integer values of r. This is what happens in particular in the high temperature phase [24,42].…”

Section: Some Remarks On the Replica Calculationmentioning

confidence: 99%

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

Derrida¹,

Mottishaw²

2015

J. Stat. Mech.

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“…In this limiting the model converges to the random energy model [23]. For recent work see [24] We are interested in the glassy behavior of the Potts model. Our approach will be to numerically solve the equation (17).…”

Section: Static Replica Equations For the Potts Glassmentioning

confidence: 99%

On the static and dynamical transition in the mean-field Potts glass

Santis

Parisi

Ritort

1995

J. Phys. A: Math. Gen.

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We study the static as well as the glassy or dynamical transition in the meanfield p-state Potts glass. By numerical solution of the saddle point equations we investigate the static and the dynamical transition for all values of p in the nonperturbative regime p > 4. The static and dynamical Edwards-Anderson parameter increase with p logarithmically. This makes the glassy transition temperature lie very close to the static one. We compare the main predictions of the theory with the numerical simulations.cond-mat/9410093 ROM2F/94/45 1

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“…For this purpose we follow the method (introduced in this context by [8,11]) of transforming the previous sum into an integral in the complex plane around the integers r = 0, 1, 2, 3...∞ and then deform the contour of integration to obtain an integral in one variable that can be evaluated by the saddle point method in the complex plane.…”

Section: The Leading Termmentioning

confidence: 99%

Replica Method and Finite Volume Corrections

2009

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In this note we introduce a method to calculate the finite volume corrections to the mean field results for the free energy when replica symmetry is broken at one-step. We find that the naive results are modified by the presence of additional corrections: these corrections can be interpreted as arising from fluctuations in the size of the blocks in the replica approach. The computation suggests a new approach for deriving the replica broken results in a rigorous way

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Some non-perturbative calculations on spin glasses

Cited by 7 publications

References 10 publications

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

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

On the static and dynamical transition in the mean-field Potts glass

Replica Method and Finite Volume Corrections

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