Analysis of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>∞</mml:mi></mml:math>-replica symmetry breaking solution of the Sherrington-Kirkpatrick model

Crisanti, A.; Rizzo, Tommaso

doi:10.1103/physreve.65.046137

Cited by 110 publications

(175 citation statements)

References 21 publications

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“…. because there is no n 6 term in Φ(n)) and ii) the coefficients of the series grow quickly with order, actually we expect the series to be asymptotic as is usually the case in this context [15].…”

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confidence: 87%

“…There are many ways in which one can compute the maximum of F n [q]. Here we follow [15,16] and introduce Lagrange multipliers P (x, y) to enforce equation (7). The resulting equations are:…”

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confidence: 99%

“…We have solved the RSB equations and computed q(n, x) and Φ(n) as a series in powers of n and τ = 1 − T [15] up to the 18th order, the series is reported in appendix. It can be proved (and it is confirmed by the explicit computation) that the lowest power of n in the expansion of Φ(n) is n 5 and that there is no n 6 term.…”

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confidence: 99%

“…In order to bypass this problem and have a good control on ∆Σ(∆e) we have adopted a method introduced in [15] to obtain q(x, τ ) from its series in powers of x and τ . We have transformed the series of ∆Σ(∆f ) in powers of ∆f and τ in a power series of just τ by setting ∆f = ( 2 45 s 5 + 1 4 τ 7 )c with c a parameter in the range [0, 1].…”

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confidence: 99%

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Large Deviations in the Free Energy of Mean-Field Spin Glasses

Parisi

Rizzo

2008

Phys. Rev. Lett.

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confidence: 87%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Large Deviations in the Free Energy of Mean-Field Spin Glasses

Parisi

Rizzo

2008

Phys. Rev. Lett.

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“…[40,41], but with an improved numerical method which allows for very accurate results for all temperatures (see Refs. [42,43]). …”

Section: A Numerical Integration Of the Frsb Equations: The Pseudo-smentioning

confidence: 99%

Thermodynamic properties of a full-replica-symmetry-breaking Ising spin glass on lattice gas: The random Blume-Emery-Griffiths-Capel model

Crisanti

Leuzzi

2004

Phys. Rev. B

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The study of the mean-field static solution of the Random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable Replica Symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case coexistence of phases takes place. The thermodynamics is worked out in the Full Replica Symmetry Breaking scheme, and the relative Parisi equations are solved using a pseudo-spectral method down to zero temperature.Since its discovery, the spin glass (SG) phase has played and still plays a fundamental role in the investigation and understanding of many basic properties of disordered and complex systems. The analysis of the mean-field approximation of theoretical models displaying such a phase has revealed different possible scenarios, including different kinds of transition from the paramagnetic phase to the SG phase, as well as different kinds of SG phases. Most of the work, however, has been concentrated on just two scenarios.In order of appearance in literature the first scenario is described by a Full Replica Symmetry Breaking (FRSB) solution characterized by a continuous order parameter function, 1 which continuously grows from zero by crossing the transition. The prototype model is the Sherrington-Kirkpatrick (SK) model, 2 a fully connected Ising-spin model with quenched random magnetic interactions.The second scenario, initially introduced by Derrida by means of the Random Energy Model (REM), 3 provides a transition with a jump in the order parameter to a stable low temperature phase in which the replica symmetry is spontaneously broken only once. The order parameter is a step function taking two values q min and the so-called Edwards-Anderson order parameter, 4 q EA , (else said self-overlap), with q min < q EA . In the paramagnetic phase they are both equal to zero. At the transition, q EA grows to a value larger than q min . No discontinuity appear, however, in the thermodynamic functions. Actually, at the transition to the one step Replica Symmetry Breaking (1RSB) SG phase, the Edwards-Anderson order parameter q EA can either grow continuously from zero or jump discontinuously to a finite value. The first case of this second scenario includes Potts-glasses with three or four states, 5 the spherical p-spin spin-glass model in strong magnetic field 6 and some inhomogeneous spherical p-spin model with a mixture of p = 2 and p > 3 interactions.7,8 The latter case includes, instead, Potts-glasses with more than four states 5 , quadrupolar glass models, 5,9 p-spin interaction spin-glass models with p...

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Some Aspects of Infinite-Range Models of Spin Glasses: Theory and Numerical Simulations

Billoire

Rugged Free Energy Landscapes

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The Edwards-Anderson Ising (EAI) model is the classical Ising model with quenched random interactions. The Hamiltonian iswhere the variables σ i = ±1 are Ising spins living on the sites of a regular square lattice in d dimensions. H is the magnetic field. The interaction involves all nearest neighbor pairs of spins, with a strength J i,j that depends on the particular link < i, j >. The J i,j 's are quenched variables, namely they do not fluctuate. They have been drawn (once for ever) independently from a unique probability distribution P (J) with mean J 0 and square deviation J 2 . In what

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Analysis of the∞-replica symmetry breaking solution of the Sherrington-Kirkpatrick model

Cited by 110 publications

References 21 publications

Large Deviations in the Free Energy of Mean-Field Spin Glasses

Large Deviations in the Free Energy of Mean-Field Spin Glasses

Thermodynamic properties of a full-replica-symmetry-breaking Ising spin glass on lattice gas: The random Blume-Emery-Griffiths-Capel model

Some Aspects of Infinite-Range Models of Spin Glasses: Theory and Numerical Simulations

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