arXiv subjects
Publications and source records attributed to Hong-Bin Chen.
We prove full replica symmetry breaking for the zero-field Sherrington-Kirkpatrick model at zero temperature: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$. At inverse temperature $\beta>1$, [arxiv.org/abs/2607.11756v3] recently proved that the Parisi measure has support $[0,q_\beta]$. Here, we show $q_\beta$ converges to $1$ as $\beta\to\infty$.
We identify the limit free energy of all multi-species spin glasses with centered $\pm 1$ spins. The result was previously known only under a convexity assumption on the covariance function of the Hamiltonian. We also obtain a one-species reduction of the formula for balanced multi-species models.
We study the full-plane two-state fuzzy $4$-Potts model, obtained by assigning independent balanced $\{\pm1\}$ spins to the open clusters of a critical $q=4$ random-cluster configuration. This model corresponds exactly to the single-spin projection of the isotropic Ashkin-Teller model at its Potts point. We prove that, after proper normalization, all even multi-point spin correlation functions converge to explicit conformally covariant Coulomb-gas type neutral charge sums. As a consequence, we prove convergence in law of the rescaled magnetization field and identify the moments of the limiting field. The proof combines the Baxter-Kelland-Wu coupling, convergence of the six-vertex height function to the Gaussian free field, and a charge-completion mechanism: an enlarged discrete sum over charge assignments with total charge in $4\mathbb Z$ produces a combinatorial cancellation of connection patterns, while only the neutral charge sector survives in the scaling limit.
Using the Baxter-Kelland-Wu coupling and the convergence of the height function of the six-vertex model to the Gaussian Free Field, we extract critical exponents for the planar critical random-cluster model at $q=4$, and the planar four-state Potts model.
The quantum kernel method, a promising quantum machine learning algorithm, possesses substantial potential for demonstrating quantum advantage. Although the majority of the quantum kernel is constructed in the context of gate-based quantum circuits, inspired by the idea of analog quantum computing, here we construct an analog quantum kernel and a hybrid quantum kernel, and show their competitiveness against other kernel methods in a benchmarking task and the practical problem of estimating non-Markovianity from sparse data. Additionally, we also incorporate operational noise into the quantum kernels. Our results reveal that the presence of operational noise can be beneficial to the performance of the developed quantum kernels. We attribute this counterintuitive noise-enhanced performance to the improved expressivity and higher model complexity induced by noise. These results pave the way for practical implementations of quantum kernel methods and provide an efficient approach for estimating non-Markovianity with reduced experimental demands.
This paper constitutes the first part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.
This paper constitutes the second part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part does not satisfy the usual convexity assumption. In this case, the Parisi formula breaks down, and there are no known methods to fully identify the limit free energy. It was suggested in [arXiv:1906.08471] that the limit free energy may be described using the unique solution of a partial differential equation of Hamilton--Jacobi type. In the present paper, we prove the validity of this conjecture in the high-temperature regime and provide an explicit representation for the free energy in terms of critical points. Using the duality between the free energy and large deviation principles, one can then easily deduce from the previous result a large deviation principle for the mean magnetization as well as a representation for the free energy of spin glass models with additional Mattis interaction at high temperature. In the companion paper, we establish similar results at all temperatures for models whose spin glass part is assumed to satisfy the usual convexity assumption.
We describe the large deviations above its typical value of the maximal energy of a spin glass with +/-1 spins. Thanks to the relatively explicit description of the rate function we identify, we then show that the latter is asymptotically quadratic near its minimum if and only if an external magnetic field is present. The proof starts from a Parisi-type formula for the fractional moments of the partition function, which we then leverage to obtain the limit of the Laplace transform of the maximum energy. Using convex-duality arguments, we then rewrite this Laplace transform as a supremum over martingales, and thereby deduce the large-deviation principle with explicit rate function.
The limit free energy of spin-glass models with convex interactions can be represented as a variational problem involving an explicit functional. Models with non-convex interactions are much less well-understood, and simple variational formulas involving the same functional are known to be invalid in general. We show here that a slightly weaker property of the limit free energy does extend to non-convex models. Indeed, under the assumption that the limit free energy exists, we show that this limit can always be represented as a critical value of the said functional. Up to a small perturbation of the parameters defining the model, we also show that any subsequential limit of the law of the overlap matrix is a critical point of this functional. We believe that these results capture the fundamental conclusions of the non-rigorous replica method.
We consider the Potts spin glass with additional ferromagnetic interaction parametrized by $t$. It has long been observed that the Potts color symmetry breaking for the spin glass order parameter is closely related to the ferromagnetic phase transition. To clarify this, we identify a single critical value $t_\mathrm{c}$, which marks the onset of both color symmetry breaking and the transition to ferromagnetism.
We consider mean-field vector spin glasses with possibly non-convex interactions. Up to a small perturbation of the parameters defining the model, the asymptotic behavior of the Gibbs measure is described in terms of a critical point of an explicit functional. In this paper, we study some properties of these critical points. Under modest assumptions ensuring that different types of spins interact, we show that the replica-symmetry-breaking structures of the different types of spins are in one-to-one correspondence with one another. For instance, if some type of spins displays one level of replica-symmetry breaking, then so do all the other types of spins. This extends the recent results of [Electronic Journal of Probability, 27:1-75, 2022] and [Comm. Math. Phys., 394(3):1101-1152, 2022] that were obtained in the case of multi-species spherical spin glasses with convex interactions.
Recently, [DOI:10.1007/s10955-023-03135-1] considered spin glass models with additional conventional order parameters characterizing single-replica properties. These parameters are distinct from the standard order parameter, the overlap, used to measure correlations between replicas. A ``min-max'' formula for the free energy was prescribed in [DOI:10.1007/s10955-023-03135-1]. We rigorously verify this prescription in the setting of vector spin glass models featuring additional deterministic spin interactions. Notably, our results can be viewed as a generalization of the Parisi formula for vector spin glass models in [DOI:10.1214/17-AOP1194], where the order parameter for self-overlap is already present.
Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results.
We study the free energy of mean-field multi-species spin glasses with convex covariance function. For such models with $D$ species, the Parisi formula is known to be valid, and expresses the limit free energy as a supremum over monotone probability measures on $\mathbb{R}_+^D$. We show here that one can transform this representation into a supremum over all probability measures on $\mathbb{R}_+^D$ of a concave functional. We then deduce that the Parisi formula admits a unique maximizer. Using convex-duality arguments, we also obtain a new representation of the free energy as an infimum over martingales in a Wiener space.
We give a meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf--Lax formula, or the Hopf formula on cones. Previously, two notions of solutions were considered, one defined directly as the Hopf--Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.
In the PDE approach to mean-field spin glasses, it has been observed that the free energy of convex spin glass models could be enriched by adding an extra parameter in its definition, and that the thermodynamic limit of the enriched free energy satisfies a partial differential equation. This parameter can be thought of as a matrix-valued path, and the usual free energy is recovered by setting this parameter to be the constant path taking only the value $0$. Furthermore, the enriched free energy can be expressed using a variational formula, which is a natural extension of the Parisi formula for the usual free energy. For models with scalar spins the Parisi formula can be expressed as an optimization problem over a convex set, and it was shown in [arXiv:1402.5132] that this problem has a unique optimizer thanks to a strict convexity property. For models with vector spins, the Parisi formula cannot easily be written as a convex optimization problem. In this paper, we generalize the uniqueness of Parisi measures proven in [arXiv:1402.5132] to the enriched free energy of models with vector spins when the extra parameter is a strictly increasing path. Our approach relies on a Gateaux differentiability property of the free energy and the envelope theorem.
In this short note, we consider models of optimal Bayesian inference of finite-rank tensor products. We add to the model a linear channel parametrized by $h$. We show that at every interior differentiable point $h$ of the free energy (associated with the model), the overlap concentrates at the gradient of the free energy and the minimum mean-square error converges to a related limit. In other words, the model is replica-symmetric at every differentiable point. At any signal-to-noise ratio, such points $h$ form a full-measure set (hence $h=0$ belongs to the closure of these points). For a sufficiently low signal-to-noise ratio, we show that every interior point is a differentiable point.
In [arXiv:2311.08980], it was shown that if the limit of the free energy in a non-convex vector spin glass model exists, it must be a critical value of a certain functional. In this work, we extend this result to multi-species spin glass models with non-convex interactions, where spins from different species may lie in distinct vector spaces. Since the species proportions may be irrational and the existence of the limit of the free energy is not generally known, non-convex multi-species models cannot be approximated by vector spin models in a straightforward manner, necessitating more careful treatment.