Finite size corrections in the Sherrington-Kirkpatrick model

Abstract

We argue that when the number of spins $N$ in the SK model is finite, the Parisi scheme can be terminated after $K$ replica-symmetry breaking steps, where $K(N)\propto N^{1/6}$. We have checked this idea by Monte Carlo simulations: we expect the typical number of peaks and features $R$ in the (non-bond averaged) Parisi overlap function $P_J(q)$ to be of order $2K(N)$, and our counting (for samples of size $N$ up to 4096 spins) gives results which are consistent with our arguments. We can estimate the leading finite size correction for any thermodynamic quantity by finding its $K$ dependence in the Parisi scheme and then replacing $K$ by K(N). Our predictions of how the Edwards-Anderson order parameter and the internal energy of the system approach their thermodynamic limit compare well with the results of our Monte Carlo simulations. The $N$-dependence of the sample-to-sample fluctuations of thermodynamic quantities can also be obtained; the total internal energy should have sample-to-sample fluctuations of order $N^{1/6}$, which is again consistent with the results of our numerical simulations.

Contribution to the conference "Viewing the World through Spin Glasses" in honour of Professor David Sherrington on the occasion of his 65th birthday, 31 August - 1 September 2007

References in corpus (25)

• Analysis of the infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model
• Energy exponents and corrections to scaling in Ising spin glasses
• Extremal Optimization for Sherrington-Kirkpatrick Spin Glasses
• Multi-Overlap Simulations of the $3d$ Edwards-Anderson Ising Spin Glass
• The Complexity of Ising Spin Glasses
• Aspect-Ratio Scaling and The Stiffness Exponent $\theta$ for Ising Spin Glasses
• Evidences Against Temperature Chaos in Mean Field and Realistic Spin Glasses
• The stiffness exponent of two-dimensional Ising spin glasses for non-periodic boundary conditions using aspect-ratio scaling
• Interface energies in Ising spin glasses
• Universality-class dependence of energy distributions in spin glasses
• Stiffness Exponents for Lattice Spin Glasses in Dimensions d=3,...,6
• Low-Temperature Excitations of Dilute Lattice Spin Glasses
• Double Criticality of the Sherrington-Kirkpatrick Model at T=0
• Hysteretic optimization for the Sherrington-Kirkpatrick spin glass
• Overlap Among States at Different Temperatures in the SK Model
• Numerical Results for Ground States of Spin Glasses on Bethe Lattices
• Spin glass overlap barriers in three and four dimensions
• Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses
• Free energy fluctuations in Ising spin glasses
• Numerical estimate of finite size corrections to the free energy of the SK model using Guerra--Toninelli interpolation
• The Stability of the Replica Symmetric State in Finite Dimensional Spin Glasses
• Nature of perturbation theory in spin glasses
• Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature
• On the Tail of the Overlap Probability Distribution in the Sherrington--Kirkpatrick Model
• Size dependence of the internal energy in Ising and vector spin glasses

Cited by in corpus (44)

• Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines
• Strong ergodicity breaking in aging of mean field spin glasses
• Evidence of non-mean-field-like low-temperature behavior in the Edwards-Anderson spin-glass model
• Eigenvalue method to compute the largest relaxation time of disordered systems
• Typical versus averaged overlap distribution in Spin-Glasses : Evidence for the droplet scaling theory
• Comment on "Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model"
• Simulations of Ground State Fluctuations in Mean-Field Ising Spin Glasses
• Finite-size scaling analysis of the distributions of pseudo-critical temperatures in spin glasses
• Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass
• One-dimensional infinite component vector spin glass with long-range interactions
• Which measures of spin-glass overlaps are informative?
• Universality and Deviations in Disordered Systems
• The cumulative overlap distribution function in realistic spin glasses
• Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model
• Analysis of the Relation between Quadratic Unconstrained Binary Optimization (QUBO) and the Spin Glass Ground-State Problem
• Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions
• Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model
• Finite-size scaling of the random-field Ising model above the upper critical dimension
• Role of the Tracy-Widom distribution in the finite-size fluctuations of the critical temperature of the Sherrington-Kirkpatrick spin glass
• Low-temperature spin-glass behaviour in a diluted dipolar Ising system
• The metastable minima of the Heisenberg spin glass in a random magnetic field
• The classical mutual information in mean-field spin glass models
• Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass
• Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
• The realizable solutions of the TAP equations
• Solving combinatorial optimization problems through stochastic Landau-Lifshitz-Gilbert dynamical systems
• Distribution of time scales in the Sherrington-Kirkpatrick model
• Finite size scaling of the de Almeida-Thouless instability in random sparse networks
• Numerical Results for Spin Glass Ground States on Bethe Lattices: Gaussian Bonds
• Phase diagram for ensembles of random close packed Ising-like dipoles as a function of texturation
• Deep reinforced learning heuristic tested on spin-glass ground states: The larger picture
• Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds
• Biases in Inverse Ising Estimates of Near-Critical Behaviour
• Pair correlation function for spin glasses
• The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
• Dealing with correlated choices: How a spin-glass model can help political parties select their policies
• Free energy barriers in the Sherrington-Kirkpatrick model
• What makes slow samples slow in the Sherrington-Kirkpatrick model
• Sample-to-sample fluctuations and bond chaos in the $m$-component spin glass
• Physics of the Edwards-Anderson Spin Glass in Dimensions $d=3,\dots ,8$ from Heuristic Ground State Optimization
• Nature of spin glass order in physical dimensions
• A replica trick for rare samples
• Ground States of the Mean-Field Spin Glass with 3-Spin Couplings
• Replica Symmetry Broken States of some Glass Models