In this paper we prove that in the high temperature region of the Sherrington-Kirkpatrick model for a typical realization of the disorder the weighted average of spins $\sum_{i\leq N} t_i σ_i$ will be approximately Gaussian provided that $\max_{i\leq N}|t_i|/\sum_{i\leq N} t_i^2$ is small.
We study the phase transition of the free energy in the Sherrington-Kirkpatrick mean-field spin glass model with centered Gaussian external field. We show that the corresponding Almeida-Thouless line is the correct transition curve that distinguishes between the replica symmetric and replica symmetry breaking solutions in the Parisi formula.
We study local optima of the Hamiltonian of the Sherrington-Kirkpatrick model. We compute the exponent of the expected number of local optima and determine the "typical" value of the Hamiltonian.
High temperature Sherrington-Kirkpatrick model for general spins
Francesco Guerra和Fabio Toninelli开发了一种非常强大的技术来研究Sherrington-Kirkpatrick平均场自旋玻璃模型的高温行为。结果表明，该模型与线性模型具有渐近可比性。关键因素是一个聪明的插值技术之间的两个不同的哈密顿描述的模型。本文的主要贡献如下：（1）复制对称解不仅适用于$\pm 1$值，而且适用于一般自旋。(2)证明过程中不涉及空化现象，只涉及一阶微分学和高斯分部积分法。
We consider a spherical spin system with pure 2-spin spherical Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank …
We prove the existence of a critical regime for the fluctuations of the ground-state energy of the spherical Sherrington-Kirkpatrick model in an external field, confirming predictions given in [3,12]. We also establish a critical regime for the fluctuations in a model with a critical Ferromagnetic interaction term, producing a three-parameter family of distributions generalizing the two-parameter family given in [4]. These results are both established in the generality of a $β$-ensemble analogue of the spherical Sherrington-Kirkpatrick model, which subsumes the complex and quarternionic genera…
We establish a strict asymptotic inequality between a class of graph partition problems on the sparse End\H{o]s-Rényi and random regular graph ensembles with the same average degree. Along the way, we establish a variational representation for the ground state energy for generalized mixed $p$-spin glasses and derive strict comparison inequalities for such models as the alphabet changes.
We consider the $2$-spin spherical Sherrington--Kirkpatrick model whose disorder is given by a deformed Wigner matrix of the form $W+λV$, where $W$ is a Wigner matrix and $V$ is a random diagonal matrix with i.i.d. entries. Assuming that the density function of the entries of $V$ decays faster than a certain rate near the edges of its spectrum, we prove the sharp phase transition of the limiting free energy and its fluctuation. In the high temperature regime, the fluctuation of $F_N$ converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low…
We consider the free energy of the bipartite spherical Sherrington--Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy--Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.
We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.
Disorder chaos in the Sherrington-Kirkpatrick model with external field
本文考虑一个自旋系统，它是由两个不同的Sherrington-Kirkpatrick（SK）模型耦合而成的，它们具有相同的温度和外场，并且它们的哈密顿量是相关的。SK模型的无序混沌猜想指出，在相应的吉布斯测度下的重叠基本上集中在一个单一的值。在没有外场的情况下，这一说法首先由Chatterjee [Disorder chaos and multiple valley in spin glasses（2009）Preprint]证实。本文利用Guerra的对称性破缺界，证明了SK模型在外场存在下也是混沌的，并且重叠的位置由一个与Guerra界和Parisi测度有关的方程决定.
We consider the quantum Sherrington-Kirkpatrick (SK) spin-glass model with transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures $β>0$. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with $\mathbb{R}^d$-valued spins. This enables us to use results of Panchenko who generalized in \cite{Pan2,Pan3} the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynam…
We study the universality of superconcentration for the free energy in the Sherrington-Kirkpatrick (SK) model. In arXiv:0907.3381, Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log{N}$, in contrast to the $O(N)$ bound obtained from the Gaussian-Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing fun…
The authors of [Ann. Henri Poincaré 16 (2015) 691-708] introduced a multi-species version of the Sherrington-Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra's replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda-Guerra identities and showing that they force the overlaps within species to be com…
We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \citet{Baiklee16} and \citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \citet{Baiklee16} in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the sph…
Let ${\boldsymbol A}\in{\mathbb R}^{n\times n}$ be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing $\langle{\boldsymbol σ},{\boldsymbol A}{\boldsymbol σ}\rangle$ over binary vectors ${\boldsymbol σ}\in\{+1,-1\}^n$. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. …
We show that, if $\mathbf{W} \in \mathbb{R}^{N \times N}_{\mathsf{sym}}$ is drawn from the gaussian orthogonal ensemble, then with high probability the degree 4 sum-of-squares relaxation cannot certify an upper bound on the objective $N^{-1} \cdot \mathbf{x}^\top \mathbf{W} \mathbf{x}$ under the constraints $x_i^2 - 1 = 0$ (i.e. $\mathbf{x} \in \{ \pm 1 \}^N$) that is asymptotically smaller than $λ_{\max}(\mathbf{W}) \approx 2$. We also conjecture a proof technique for lower bounds against sum-of-squares relaxations of any degree held constant as $N \to \infty$, by proposing an approximate pse…
