From Zero-Freeness to Strong Spatial Mixing for the General 2-Spin System

Conceitos Básicos

Zero-freeness of the partition function of the general 2-spin system implies strong spatial mixing on the corresponding parameter regions.

Resumo

The paper presents a unifying approach to derive the strong spatial mixing (SSM) property for the general 2-spin system from zero-free regions of its partition function. The approach works for the multivariate partition function over all three complex parameters (β, γ, λ), and allows the zero-free regions of β, γ or λ to be of arbitrary shapes, as long as they contain certain specific points.
The key technical contribution is the establishment of a Christoffel-Darboux type identity for the 2-spin system on trees. This identity plays an important role in the approach and enables the authors to prove that zero-freeness implies two key properties: local dependence of coefficients (LDC) and uniform bound on a circle. These two properties are then used to show that zero-freeness implies SSM.
The approach comprehensively turns all existing zero-free regions of the 2-spin system partition function (where pinned vertices are allowed) into the SSM property. As a consequence, new SSM results are obtained beyond the direct argument for SSM based on tree recurrence. The approach is also extended to handle the 2-spin system with non-uniform external fields, leading to a new SSM result for the non-uniform ferromagnetic Ising model from the Lee-Yang circle theorem.

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Principais Insights Extraídos De

From Zero-Freeness to Strong Spatial Mixing via a Christoffel-Darboux Type Identity

by Shuai Shao,X... às arxiv.org 04-11-2024

https://arxiv.org/pdf/2401.09317.pdf

Perguntas Mais Profundas

How can the approach be extended to derive SSM from zero-freeness in other complex parameter regions beyond the ones covered in this paper

The approach presented in the paper can be extended to derive Strong Spatial Mixing (SSM) from zero-freeness in other complex parameter regions by considering different combinations of the parameters (β, γ, λ) and analyzing the zero-free regions for each combination. This extension would involve exploring the behavior of the partition function for the 2-spin system on various graphs under different complex parameter settings. By identifying additional zero-free regions where the partition function does not vanish, the approach can be applied to show that SSM holds for the 2-spin system in those regions. This extension would require a thorough analysis of the properties of the partition function and the relationships between the parameters to establish the conditions under which SSM can be derived from zero-freeness in these new complex parameter regions.

Can the Christoffel-Darboux type identity established in this paper be useful in other contexts beyond deriving SSM from zero-freeness

The Christoffel-Darboux type identity established in this paper can be useful in other contexts beyond deriving SSM from zero-freeness. This identity provides a fundamental relationship between the partition functions of the 2-spin system with different pinned vertices on trees. Such an identity can be valuable in studying the properties of the 2-spin system, analyzing the behavior of the partition function under various conditions, and understanding the interactions between the parameters. Additionally, the Christoffel-Darboux type identity can potentially be applied in other areas of mathematical analysis, graph theory, and statistical physics where similar relationships between functions or systems need to be established.

Are there other interesting models for which SSM-based FPTASes can be obtained from zero-freeness via the approach presented in this paper

There are several other interesting models for which Strong Spatial Mixing (SSM)-based Fully Polynomial-Time Approximation Schemes (FPTASes) can potentially be obtained from zero-freeness using the approach presented in this paper. One such model is the Potts model, which generalizes the Ising model to include multiple spin states. By applying the same principles of zero-freeness and SSM to the partition function of the Potts model with complex parameters, it may be possible to derive FPTASes for this model as well. Additionally, other spin systems, such as the XY model or the Heisenberg model, could also be explored using this approach to investigate the relationship between zero-freeness and SSM and potentially develop efficient approximation schemes for these models.