The Cavity Method at Zero Temperature

  title={The Cavity Method at Zero Temperature},
  author={Marc M{\'e}zard and Giorgio Parisi},
  journal={Journal of Statistical Physics},
In this note we explain the use of the cavity method directly at zero temperature, in the case of the spin glass on a lattice with a local tree like structure, which is the proper generalization of the usual Bethe lattice to frustrated problems. The computation is done explicitly in the formalism equivalent to “one step replica symmetry breaking;” we compute the energy of the global ground state, as well as the complexity of equilibrium states at a given energy. Full results are presented for a… 
Replica Cluster Variational Method
We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi’s Cluster Variational Method (CVM). Using replicas and the message-passing
Role of fluctuations in the phase transitions of coupled plaquette spin models of glasses
We study the role of fluctuations on the thermodynamic glassy properties of plaquette spin models, more specifically on the transition involving an overlap order parameter in the presence of an
Near-optimal configurations in mean-field disordered systems.
We present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems. Our approach is based on a
Replica symmetry breaking in the 'small world' spin glass
We apply the cavity method to an Ising spin glass model on a 'small world' lattice, a random bond graph superimposed upon a one-dimensional ferromagnetic ring, which allows us to identify the cavity
Spin glass models with ferromagnetically biased couplings on the Bethe lattice: analytic solutions and numerical simulations
Abstract.We derive the zero-temperature phase diagram of spin glass models with a generic fraction of ferromagnetic interactions on the Bethe lattice. We use the cavity method at the level of
Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model).
Numerical results for spin glass ground states on Bethe lattices: Gaussian bonds
AbstractThe average ground state energies for spin glasses on Bethe lattices of connectivities r = 3,...,15 are studied numerically for a Gaussian bond distribution. The Extremal Optimization
The Marginally Stable Bethe Lattice Spin Glass Revisited
Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a
Numerical results for ground states of spin glasses on Bethe lattices
Abstract:The average ground state energy and entropy for ±J spin glasses on Bethe lattices of connectivities k + 1 = 3..., 26 at T = 0 are approximated numerically. To obtain sufficient accuracy for
Zero temperature solutions of the Edwards-Anderson model in random Husimi lattices
We solve the Edwards-Anderson model (EA) in different Husimi lattices using the cavity method at replica symmetric (RS) and 1-step of replica symmetry breaking (1RSB) levels. We show that, at T = 0,


Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick
The Bethe lattice spin glass revisited
Abstract:So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties,
A mean field spin glass with short-range interactions
We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties.
Spin-glass on a Bethe lattice.
  • M. Thouless
  • Physics, Medicine
    Physical review letters
  • 1986
The Ising spin-glass in a magnetic field is studied for the Bethe lattice. There is an instability that agrees with the replica-symmetry-breaking transition found for the infinite-range model.
Spin-glass theory in the Bethe approximation: Insights and problems
We set up and solve the Bethe approximation for a spin-glass with finite-range interactions. We obtain the finite-$z$ analog of the Thouless-Anderson-Palmer equations and the free-energy functional
Two- and three-spin cluster theory of spin-glasses
The cluster effect of spin-glasses is studied on the basis of the two- and three-spin cluster theory. In contrast to ordinary ferromagnets, the effective field containing the cluster effect is shown
Bethe lattice spin glass: The effects of a ferromagnetic bias and external fields. I. Bifurcation analysis
We present a rigorous analysis of the ±J Ising spin-glass model on the Bethe lattice with fixed uncorrelated boundary conditions. Phase diagrams are derived as a function of temperature vs.
Monte Carlo simulations of the Ising spin glass on lattices with finite connectivity
The authors have simulated the Ising spin-glass model on a random lattice with a finite (average) coordination number and also on the Bethe lattice with various different boundary conditions. In
From inherent structures to pure states: Some simple remarks and examples
The notions of pure states and inherent structures, i.e. stable configurations against 1-spin flip are discussed. We explain why these different concepts accidentally coincide in mean-field models
Exact solutions for diluted spin glasses and optimization problems
Using a one-step functional replica symmetry breaking ansatz, the resulting ground state energy is in perfect agreement with numerical simulations of p-spin glass models with finite connectivity and of some optimization problems.