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

一、题目：From Zero-Freeness to Strong Spatial Mixing via a Christoffel-Darboux Type Identity

二、主讲人：邵帅 教授（中国科学技术大学）

三、摘要

Abstract: In this talk, we show how to derive the strong spatial mixing property for 2-spin systems from zero-free regions of its partition function. We view the partition function of 2-spin systems as a multivariate function over three complex variables (β,γ,λ), and we allow the zero-free regions of β,γ or λ to be of arbitrary shapes. As long as the zero-free region contains a positive point and it is a complex neighborhood of λ = 0 when fixing β,γ ∈ C , or a complex neighborhood of βγ = 1 when fixing β,λ ∈ C or γ,λ ∈ C, we are able to show that the corresponding 2-spin system exhibits strong spatial mixing on such a region. The underlying graphs of the 2-spin system are not necessarily of bounded degree, while are required to include graphs with pinned vertices. We prove this result by establishing a Christoffel-Darboux type identity for the 2-spin system on trees and using certain tools from complex analysis.

To our best knowledge, our result is general enough to turn all currently known zero-free regions of the partition function of 2-spin systems where pinned vertices are allowed into the strong spatial mixing property. Moreover, we apply our result to obtain strong spatial mixing for the ferromagnetic Ising model from the celebrated Lee-Yang circle theorem.

Based on joint work with Ke Shi and Xiaowei Ye.

四、主讲人简介

邵帅，中国科大计算机学院特任教授。2014年本科毕业于中国科大少年班学院华罗庚班，2020年博士毕业于威斯康星大学麦迪逊分校计算机系；之后分别在牛津大学及爱丁堡大学从事博士后工作，在牛津工作期间同时被选为Wolfson学院初级研究员。研究领域为理论计算机科学，主要研究问题包括Holant（即edge-CSP）框架下判定、优化及计数问题的计算复杂性分类，同时涉及其与统计物理中相变概念以及量⼦纠缠理论的交叉问题。

五、邀请人：张鹏

六、时间：2024年4月1日（星期一）14:00-16:00

七、地点：软件学院办公楼310会议室

八、主办：山东大学软件学院