Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

Sharp Signal Detection Under Ferromagnetic Ising Models

In this paper we study the effect of dependence on detecting a class of structured signals in Ferromagnetic Ising models. Natural examples of our class include Ising Models on lattices, and Mean-Field type Ising Models such as dense Erdős-Rényi, and dense random regular graphs. Our results not only provide sharp constants of detection in each of these cases and thereby pinpoint the precise relationship of the detection problem with the underlying dependence, but also demonstrate how to be agnostic over the strength of dependence present in the respective models.

Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, Sly '11], under an appropriate "continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with $+$ and $-$ boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, …

Poisson可表示过程的混合性质及其对Ising模型和接触过程的影响

Mixing for Poisson representable processes and consequences for the Ising model and the contact process

本文为Poisson可表示过程建立新的混合特征，证明超临界接触过程上不变测度及相变区Ising模型加态等非Poisson可表示，并推广至非极值平移不变态，回应了既有问题。

Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields

We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show: (1) polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018); (2) near linear mixing time for the two dynamics above if the maxi…

Particle Systems arising from an anti-ferromagnetic Ising model

We study a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model has a bijection with a one-dimensional particle system equipped with creation and annihilation. In the thermodynamic limit, we determine the explicit phase diagrams as functions of temperature and anisotropy. Two values of the anisotropy are of particular interest - the 'critical' value and the 'independent' value. At independence, the particle system has the same distribution as the two colored noisy voter model. Its limiting measure under a natural scaling window is…

Homogenization of ferromagnetic energies on Poisson random sets in the plane

We prove that by scaling nearest-neighbour ferromagnetic energies defined on Poisson random sets in the plane we obtain an isotropic perimeter energy with a surface tension characterised by an asymptotic formula. The result relies on proving that cells with `very long' or `very short' edges of the corresponding Voronoi tessellation can be neglected. In this way we may apply Geometry Measure Theory tools to define a compact convergence, and a characterisation of metric properties of clusters of Voronoi cells using limit theorems for subadditive processes.

The critical 1-arm exponent for the ferromagnetic Ising model on the Bethe lattice

We consider the ferromagnetic nearest-neighbor Ising model on regular trees (Bethe lattice), which is well-known to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be described in terms of various critical exponents; one of them is the critical 1-arm exponent $ρ$, which characterizes the rate of decay of the (root) magnetization. The crucial quantity we analyze in this work is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related…

一个用于零场铁磁伊辛模型的 $\widetilde{O}\left(n^3 \right)$ 时间采样器

An $\widetilde{O}\left(n^2 \right)$-Time Sampler for Zero-Field Ferromagnetic Ising Models

本文提出一个在任意图上近似采样零场铁磁伊辛模型的算法，运行时间为 $\widetilde O(m+n)+\widetilde O_\beta(n^3\log^3(1/\epsilon))$，利用割稀疏化器和传输流思想，主要证明由 GPT-5.6 Sol Pro 辅助完成。

随机图上反铁磁 Ising 模型的采样困难性

Hardness of sampling for the anti-ferromagnetic Ising model on random graphs

本文证明大平均度随机图上反铁磁 Ising 模型在 β>1 时不存在接近 Gibbs 测度的稳定采样算法，并通过比较平均能量与重叠的插值方法揭示搜索与采样的可处理性分离。

Approach to Fixation for Zero-Temperature Stochastic Ising Models on the Hexagonal Lattice

We investigate zero-temperature dynamics on the hexagonal lattice H for the homogeneous ferromagnetic Ising model with zero external magnetic field and a disordered ferromagnetic Ising model with a positive external magnetic field h. We consider both continuous time (asynchronous) processes and, in the homogeneous case, also discrete time synchronous dynamics (i.e., a deterministic cellular automaton), alternating between two sublattices of H. The state space consists of assignments of -1 or +1 to each site of H, and the processes are zero-temperature limits of stochastic Ising ferromagnets wi…

在随机图上固定磁化Ising模型直至重构

Fixed-magnetization Ising on random graphs up to reconstruction

该研究在随机图上分析固定磁化Ising模型，证明其自由能收敛到退火自由能，并应用结果验证了最大割和最小二分割猜想，同时确定了大偏差率函数及Glauber动力学混合时间。

Approximating Pairwise Correlations in the Ising Model

In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approximation to this covariance by repeatedly sampling from the relevant Gibbs distribution. However, we desire a multiplicative approximation, and it is not clear how to achieve this by sampling, given that the covariance can be exponentially small. Our main contribution is a fully polynomial time randomised approximation scheme (FPRAS) for the covariance. We also show that that the restriction to the ferromagnetic case i…