Dmitry Panchenko
Recently Michel Talagrand gave a rigorous proof of the Parisi formula in the Sherrington-Kirkpatrick model. In this paper we build upon the methodology developed by Talagrand and extend his result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.
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This paper has been withdrawn since a trivial proof of the result has been pointed out to the author.
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.
In a region above the Almeida-Thouless line, where we are able to control the thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are Gaussian, on the scale N^{-1/2}, for N large. The method we employ is based on the idea, we recently developed, of introducing quadratic coupling between two replicas. The proof makes use of the cavity equations and of concentration of measure inequalities for the free energy.
We show that the limiting free energy in Sherrington-Kirkpatrick's Spin Glass Model does not depend on the environment.
We develop a simple method to study the high temperature, or high external field, behavior of the Sherrington-Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a lin…
We propose an iterative construction of solutions of the Thouless-Anderson-Palmer-equations for the Sherrington-Kirpatrick model. The iterative scheme is proved to converge exactly up to the de Almayda-Thouless-line. No results on the SK-model itself are derived.
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.
Sampling from the Sherrington-Kirkpatrick Model up to Criticality
研究针对谢林顿-柯克帕特里克模型，证明模拟退火算法在β<1时可高效采样至任意逆多项式精度，为相关统计物理与计算问题提供了理论支撑。
In this paper we consider a system of spins that consists of two configurations $\vsi^1,\vsi^2\inΣ_N=\{-1,+1\}^N$ with Gaussian Hamiltonians $H_N^1(\vsi^1)$ and $H_N^2(\vsi^2)$ correspondingly, and these configurations are coupled on the set where their overlap is fixed $\{R_{1,2}=N^{-1}\sum_{i=1}^N σ_i^1σ_i^2 = u_N\}.$ We prove the existence of the thermodynamic limit of the free energy of this system given that $\lim_{N\to\infty}u_N = u\in[-1,1]$ and give the analogue of the Aizenman-Sims-Starr variational principle that describes this limit via random overlap structures.
The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in [31] and [44].
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.
Free energy fluctuation of soft spherical Sherrington-Kirkpatrick model
本文研究软球Sherrington-Kirkpatrick模型，证明其自由能在高温区呈高斯涨落，低温区由最大特征值主导并收敛于GOE Tracy-Widom分布，表明软径向约束保留经典模型的涨落转变。
Francesco Guerra and Fabio Toninelli have developped a very powerful technique to study the high temperature behaviour of the Sherrington-Kirkpatrick mean field spin glass model. They show that this model is asymptoticaly comparable to a linear model. The key ingredient is a clever interpolation technique between the two different Hamiltonians describing the models. This paper contribution to the subject are the following: (1) The replica-symmetric solution holds for general spins, not just $\pm 1$ valued. (2) The proof does not involve cavitation but only first order differential calculus and…
We present extended versions and give detailed proofs of results concerning percolation (using various sets of two-replica bond occupation variables) in Sherrington-Kirkpatrick spin glasses (with zero external field) that were first given in an earlier paper by the same authors. We also explain how ultrametricity is manifested by the densities of large percolating clusters. Our main theorems concern the connection between these densities and the usual spin overlap distribution. Their corollaries are that the ordered spin glass phase is characterized by a unique percolating cluster of maximal d…
We develop a cavity method in the spherical Sherrington-Kirkpatrick model at high temperature and small external field. As one application we compute the limit of the covariance matrix for fluctuations of the overlap and magnetization.
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 …
Fluctuations of the ground state of the spiked spherical Sherrington-Kirkpatrick model
本文研究了带外场或非线性“尖刺”项的 Sherrington-Kirkpatrick 哈密顿量基态的最大值，计算了其领先阶水平及一阶、二阶涨落，并推导了模型 TAP 自由能最大值的等价结果。
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…
Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with sparse interaction
本文研究了具有稀疏相互作用的球面 Sherrington-Kirkpatrick 自旋玻璃模型，证明了极限自由能不依赖稀疏性，而自由能涨落在高低温下均依分布收敛到同阶但方差不同的Gaussian分布。
On étudie la covariance (pour la mesure de Gibbs) des spins en deux sites dans le cas d'un modèle de Sherrington--Kirkpatrick avec champ externe; lorsque le nombre de sites du modèle tend vers l'infini, une évaluation asymptotique des moments d'ordre $p$ de cette covariance permet d'obtenir un théorème limite faible avec une loi limite en général non gaussienne. We study the covariance (for Gibbs measure) of spins at two sites in the case of a Sherrington--Kirkpatrick model with an external field. When the number of sites of the model grows to infinity, an asymptotic evaluation of the $p$ mome…
Parisi measure of the Sherrington-Kirkpatrick model with external field
本文证明Sherrington-Kirkpatrick模型在正温度和正外场下Parisi测度支撑连通，并刻画复制对称破缺时的区间支撑、光滑密度及端点原子性质。
One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature regime of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient approach for obtaining the limiting distributions, the nature of this method restricts the derivation of optimal and general results. In this paper, our study ba…
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.
Existence of Full Replica Symmetry Breaking for the Sherrington-Kirkpatrick Model at Low Temperature
该研究证明低温下SK模型存在完全复制对称破缺，指出略高于临界温度时其Parisi测度由原点起始区间支撑且仅右端点有跳跃间断点，为该模型的低温性质提供理论依据。
The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited
研究临界逆温度β=1下Sherrington-Kirkpatrick模型自由能的波动，证明方差上界为(1/4)log N+C，下界为(1/2)log log log N-C。
Full replica symmetry breaking in the Sherrington-Kirkpatrick model
研究谢林顿 - 柯克帕特里克模型在零外场及β>1时帕里西测度的支撑情况，通过反证法，利用高斯科尔 - 霍普夫表示导出微分不等式，证明此时帕里西测度具有无限支撑。
In this paper we consider central limit theorems for various macroscopic observables in the high temperature region of the Sherrington-Kirkpatrick spin glass model. With a particular focus on obtaining a quenched central limit theorem for the energy density of the system with non-zero external field, we show how to combine the mean field cavity method with Stein's method in the quenched regime. The result for the energy density extends the corresponding result of Comets-Neveu.
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.
Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model
本文针对Sherrington-Kirkpatrick模型，利用Parisi测度的区间支撑，建立了定量Parisi公式，并证明了配分函数方差介于$N^{4/15}$与$N^{7/15}$阶，同时给出自由能和基态能量的偏差界及指数$6/5$的上尾估计。
The spherical Sherrington-Kirkpatrick model is a spherical mean field model for spin glass. We consider the fluctuations of the free energy at arbitrary non-critical temperature for the 2-spin model with no magnetic field. We show that in the high temperature regime the law of the fluctuations converges to the Gaussian distribution just like in the Sherrington-Kirtkpatrick model. We show, on the other hand, that the law of the fluctuations is given by the GOE Tracy-Widom distribution in the low temperature regime. The orders of the fluctuations are markedly different in these two regimes. A un…
We explore the joint behavior of a finite number of multi-overlaps in the high temperature phase of the SK model. Extending work by M. Talagrand, we show that, when these objects are scaled to have non-trivial limiting distributions, the joint behavior is described by a Gaussian process with an explicit covariance structure.
We study a spin system with both mixed even-spin Sherrington-Kirkpatrick (SK) couplings and Curie-Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda-Guerra identities hold on a dense subset of the temperature parameters. (iii) We establish a general inequality between the magnetization and overlap. (iv) We construct a temperat…
Replica symmetry up to the de Almeida-Thouless line in the Sherrington-Kirkpatrick model
研究在逆温度β和均匀外场h>0下，Sherrington-Kirkpatrick模型中副本对称性的成立条件，证明了在β²E[sech⁴(β√qZ+h)]≤1的范围内成立，验证了de Almeida和Thouless的预测。
Thouless-Anderson-Palmer equations for the Multi-species Sherrington-Kirkpatrick model
本文证明多种群 Sherrington-Kirkpatrick 自旋玻璃模型的局部磁化 TAP 方程，借助已有集中性结果，将高温有效性推广到方差剖面矩阵无需半正定的一般情形。
Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with heavy-tailed interaction
该研究针对无外场的重尾相互作用2自旋球形Sherrington-Kirkpatrick模型，证明自由能存在依赖相互作用矩阵最大特征值位置的尖锐相变，并得到两种区域下极限自由能的阶数与自由能涨落的极限分布。
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.
We consider a spin system obtained by coupling two distinct Sherrington-Kirkpatrick (SK) models with the same temperature and external field whose Hamiltonians are correlated. The disorder chaos conjecture for the SK model states that the overlap under the corresponding Gibbs measure is essentially concentrated at a single value. In the absence of external field, this statement was first confirmed by Chatterjee [Disorder chaos and multiple valleys in spin glasses (2009) Preprint]. In the present paper, using Guerra's replica symmetry breaking bound, we prove that the SK model is also chaotic i…
Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.
A short proof of free energy limit of two spin spherical Sherrington-Kirkpatrick model at any temperature
本文基于Laplace积分法与空间离散近似，给出了无外场球面2自旋Sherrington-Kirkpatrick模型在任意温度下自由能极限的简短证明，并推广至具有良好特征值分布性质的任意相互作用矩阵。
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…
Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature
本文研究了Sherrington-Kirkpatrick自旋玻璃模型在临界温度附近的波动性质，推导了对数分区函数的方差表达式，并证明了相关的中心极限定理。
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…
Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
本文证明Sherrington-Kirkpatrick模型在临界点的自由能方差与自旋玻璃磁化率的对数成正比，系数由磁化率标度指数决定，并利用精确恒等式得到尖锐系数1/6。
Bounds on the covariance matrix of the Sherrington-Kirkpatrick model
本文通过微分TAP方程建立近似公式并控制误差项，证明无外场且逆温度$β<1$的Sherrington-Kirkpatrick模型Gibbs测度协方差矩阵期望算子范数被仅依赖于$β$的常数界定，回答了Talagrand的开放问题，并给出临界与低温下发散下界。
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…
Weak Poincaré Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model
研究通过先证明弱泛函不等式对逼近随机定位过程的正则分布序列成立，再转移到所需分布，证明了\(\beta < \frac{1}{2}\)时谢林顿 - 柯克帕特里克模型吉布斯测度的弱庞加莱不等式，使热启动格劳伯动力学可有效抽样，向解决相关猜想迈进。
Fluctuations of the free energy of the Sherrington-Kirkpatrick model with ferromagnetic interaction
该研究针对带铁磁相互作用的Sherrington-Kirkpatrick模型，推导其铁磁区自由能的中心极限定理，完善纯2自旋模型的相图相关涨落结论，是球形SKFI模型涨落结果的伊辛类比。
Analyzing dynamics and average case complexity in the spherical Sherrington-Kirkpatrick model: a focus on extreme eigenvectors
本文研究球形 Sherrington-Kirkpatrick 模型的 Langevin 动力学与渐近能量极限，结合积分微分方程分析相变，并对 Wigner 矩阵极值特征向量及幂迭代算法进行平均情形复杂度分析，确立命中时间与复杂度界限。
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…
Universality of the fluctuations of the free energy in generalized Sherrington-Kirkpatrick models and the log likelihood ratio in spiked Wigner models
研究高温/次临界区域下广义Sherrington-Kirkpatrick模型自由能涨落和尖峰Wigner模型对数似然比的极限分布，证明在适当条件下其极限为高斯分布且具有普适性。
Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model
本文建立Ising Sherrington--Kirkpatrick模型临界窗口内自由能波动的极限分布，给出GOE结果及中等超临界区域的Tracy--Widom极限，并证明对前三阶矩匹配高斯且第四阶矩有界的独立无序矩阵具有普适性。
TAP variational principle for the constrained overlap multiple spherical Sherrington-Kirkpatrick model
本文研究了约束重叠多重球面 Sherrington-Kirkpatrick 模型，证明其自由能可用 TAP 变分原理表达，并在所有温度和外部场下实现了数学严格的 TAP 计算，同时证实了模型的副本对称性。
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…
The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model
本文研究了自交叠修正的量子谢尔林-基普里亚克模型在垂直磁场中的玻璃转变，确定了玻璃相与顺磁相之间的相界，并基于简化后的帕里斯变分原理进行证明。
Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
研究零外场临界温度β=1时Sherrington-Kirkpatrick自旋玻璃模型，证明自由能方差为(1/6)log N + O(1)并满足高斯CLT，同时确认临界两副本重叠尺度为N^{-2/3}，关键输入是Ising与球面SK配分函数之比在L^2中趋于1。
Replica symmetry breaking in multi-species Sherrington-Kirkpatrick model
本文针对多种类Sherrington-Kirkpatrick模型，在两种类正定方差结构下导出了显式的AT温度阈值，首次给出多种类自旋玻璃的非渐近对称性破缺条件，并证明其与副本对称临界点唯一性条件相吻合。
Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization
研究高温无外场Sherrington-Kirkpatrick模型的Gibbs分布采样问题，提出基于算法随机定位的$O(n^2)$复杂度采样方法，将逆温度$\beta$的有效采样范围从$1/4$提升至$1/2$，并证明$\beta>1$时无稳定算法可近似采样。
Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at $β< 1/2$
该研究针对β<1/2的Sherrington-Kirkpatrick模型，利用Bakry-Émery判据等建立最优阶庞加莱不等式，结合定位方案框架得到单站点Glauber动力学的最优混合时间界，核心思想来自GPT-5.6 Sol Ultra。
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