Determining the Shannon capacity of graphs is a long-standing open problem in information theory, graph theory and combinatorial optimization. Over decades, a wide range of upper and lower bound methods have been developed to analyze this problem. However, despite tremendous effort, even small instances of the problem have remained open. In recent years, a new dual characterization of the Shannon capacity of graphs, asymptotic spectrum duality, has unified and extended known upper bound methods and structural theorems. In this paper, building on asymptotic spectrum duality, we develop a new theory of graph distance, that we call asymptotic spectrum distance, and corresponding limits (reminiscent of, but different from, the celebrated theory of cut-norm, graphons and flag algebras). We propose a graph limit approach to the Shannon capacity problem: to determine the Shannon capacity of a graph, construct a sequence of easier to analyse graphs converging to it. (1) We give a very general construction of non-trivial converging sequences of graphs (in a family of circulant graphs). (2) We construct Cauchy sequences of finite graphs that do not converge to any finite graph, but do converge to an infinite graph. We establish strong connections between convergence questions of finite graphs and the asymptotic properties of Borsuk-like infinite graphs on the circle. (3) We observe that all best-known lower bound constructions for Shannon capacity of small odd cycles can be obtained from a "finite" version of the graph limit approach. We develop computational and theoretical aspects of this approach and use these to obtain a new Shannon capacity lower bound for the fifteen-cycle. The theory of asymptotic spectrum distance applies not only to Shannon capacity of graphs; indeed, we will develop it for a general class of mathematical objects and their asymptotic properties.

翻译：确定图的香农容量是信息论、图论与组合优化领域一个长期悬而未决的开放问题。数十年来，学界发展出多种上下界方法来分析该问题，但即便针对小规模实例，相关研究仍因巨大挑战而未能取得突破。近年来，香农容量的新对偶刻画——渐近谱对偶理论——统一并拓展了已知的上界方法与结构定理。本文基于渐近谱对偶理论，发展了一种新的图距离理论（称为渐近谱距离）及其对应的极限概念（这一概念虽令人联想起著名的割范数、图元与旗代数理论，但与之存在本质区别）。我们提出一种解决香农容量问题的图极限方法：要确定某图的香农容量，需构造一列更易分析的图序列，使其收敛于目标图。（1）我们给出循环图族中非平凡收敛图序列的普适构造方法；（2）构造了不收敛于任何有限图、但收敛于无限图的有限图柯西序列，揭示有限图收敛问题与圆上博尔苏克型无限图渐近性质之间的深刻联系；（3）发现所有已知的小奇循环香农容量下界构造均可通过图极限方法的"有限化"版本实现。我们发展了该方法的计算与理论框架，并利用其获得了十五循环香农容量的新下界。值得注意的是，渐近谱距离理论不仅适用于图香农容量问题——我们将在更一般的数学对象范畴及其渐近性质框架下系统阐述该理论。