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Register a new userThe replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings
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We study the mean-field Ising spin glass model with external field, where the random symmetric couplings matrix is orthogonally invariant in law. For sufficiently high temperature, we prove that the replica-symmetric prediction is correct for the first-order limit of the free energy. Our analysis is an adaption of a conditional quenched equals annealed argument used by Bolthausen to analyze the high-temperature regime of the Sherrington-Kirkpatrick model. We condition on a sigma-field that is generated by the iterates of an Approximate Message Passing algorithm for solving the TAP equations in this model, whose rigorous state evolution was recently established.
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We compute the high-dimensional limit of the free energy associated with a multi-layer generalized linear model. Under certain technical assumptions, we identify the limit in terms of a variational formula. The approach is to first show that the limit is a solution to a Hamilton-Jacobi equation whose initial condition is related to the limiting free energy of a model with one fewer layer. Then, we conclude by an iteration.
Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper we consider the case of Ising mixed $p$-spin models, namely Hamiltonians $H_N:Sigma_Nto {mathbb R}$ on the Hamming hypercube $Sigma_N = {pm 1}^N$, which are defined by the property that ${H_N({boldsymbol sigma})}_{{boldsymbol sigma}in Sigma_N}$ is a centered Gaussian process with covariance ${mathbb E}{H_N({boldsymbol sigma}_1)H_N({boldsymbol sigma}_2)}$ depending only on the scalar product $langle {boldsymbol sigma}_1,{boldsymbol sigma}_2rangle$. The asymptotic value of the optimum $max_{{boldsymbol sigma}in Sigma_N}H_N({boldsymbol sigma})$ was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here we ask whether a near optimal configuration ${boldsymbol sigma}$ can be computed in polynomial time. We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of $H_N$, and characterize the typical energy value it achieves. When the $p$-spin model $H_N$ satisfies a certain no-overlap gap assumption, for any $varepsilon>0$, the algorithm outputs ${boldsymbol sigma}inSigma_N$ such that $H_N({boldsymbol sigma})ge (1-varepsilon)max_{{boldsymbol sigma}} H_N({boldsymbol sigma})$, with high probability. The number of iterations is bounded in $N$ and depends uniquely on $varepsilon$. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.
We consider the free energy difference restricted to a finite volume for certain pairs of incongruent thermodynamic states (if they exist) in the Edwards-Anderson Ising spin glass at nonzero temperature. We prove that the variance of this quantity with respect to the couplings grows proportionally to the volume in any dimension greater than or equal to two. As an illustration of potential applications, we use this result to restrict the possible structure of Gibbs states in two dimensions.
Parisis formal replica-symmetry--breaking (RSB) scheme for mean-field spin glasses has long been interpreted in terms of many pure states organized ultrametrically. However, the early version of this interpretation, as applied to the short-range Edwards-Anderson model, runs into problems because as shown by Newman and Stein (NS) it does not allow for chaotic size dependence, and predicts non-self-averaging that cannot occur. NS proposed the concept of the metastate (a probability distribution over infinite-size Gibbs states in a given sample that captures the effects of chaotic size dependence) and a non-standard interpretation of the RSB results in which the metastate is non-trivial and is responsible for what was called non-self-averaging. Here we use the effective field theory of RSB, in conjunction with the rigorous definitions of pure states and the metastate in infinite-size systems, to show that the non-standard picture follows directly from the RSB mean-field theory. In addition, the metastate-averaged state possesses power-law correlations throughout the low temperature phase; the corresponding exponent $zeta$ takes the value $4$ according to the field theory in high dimensions $d$, and describes the effective fractal dimension of clusters of spins. Further, the logarithm of the number of pure states in the decomposition of the metastate-averaged state that can be distinguished if only correlations in a window of size $W$ can be observed is of order $W^{d-zeta}$. These results extend the non-standard picture quantitatively; we show that arguments against this scenario are inconclusive.
The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated, with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-paramagnetic phase boundary is probed, with the following estimates for the associated critical indices: $ u=1.49(2)$, $gamma=2.71(4)$, $eta_1= 0.183(3)$, distinctly different from the percolation values.
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