计算规则图形的高斯积分

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Oleg Evnin, Weerawit Horinouchi
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引用次数: 0

摘要

在最近的一篇文章[Kawamoto, J. Phys. Complexity 4, 035005 (2023)]中,川本引用了统计物理学方法来解决具有规定度序列的图计数问题。这种处理方法涉及截断特定泰勒展开的前两项,从而得出图形计数的本德尔-坎菲尔德估计值。这种方法出乎意料地成功,因为本德尔-坎菲尔德公式对于大型图来说是渐进精确的,而数列截断法并没有先验地显示出类似的精确度。我们从三个方面提升了这一处理方法。首先,我们用 d 维高斯积分推导出计算 d 不规则图的精确公式。其次,我们展示了如何将该公式转换为 d 不规则图计数生成函数的积分表示。第三,我们对大图形尺寸进行了明确的鞍点分析,并确定了对图形计数估计负责的鞍点配置。在这些鞍点配置中,只有两个积分变量浓缩为重要值,而其余变量在大型图中趋近于零。这提供了一个基本图景,证明了川本早先的发现是正确的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Gaussian integral that counts regular graphs
In a recent article [Kawamoto, J. Phys. Complexity 4, 035005 (2023)], Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the first two terms, which resulted in the Bender-Canfield estimate for the graph counts. This is surprisingly successful since the Bender-Canfield formula is asymptotically accurate for large graphs, while the series truncation does not a priori suggest a similar level of accuracy. We upgrade this treatment in three directions. First, we derive an exact formula for counting d-regular graphs in terms of a d-dimensional Gaussian integral. Second, we show how to convert this formula into an integral representation for the generating function of d-regular graph counts. Third, we perform explicit saddle point analysis for large graph sizes and identify the saddle point configurations responsible for graph count estimates. In these saddle point configurations, only two of the integration variables condense to significant values, while the remaining ones approach zero for large graphs. This provides an underlying picture that justifies Kawamoto’s earlier findings.
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来源期刊
Journal of Mathematical Physics
Journal of Mathematical Physics 物理-物理:数学物理
CiteScore
2.20
自引率
15.40%
发文量
396
审稿时长
4.3 months
期刊介绍: Since 1960, the Journal of Mathematical Physics (JMP) has published some of the best papers from outstanding mathematicians and physicists. JMP was the first journal in the field of mathematical physics and publishes research that connects the application of mathematics to problems in physics, as well as illustrates the development of mathematical methods for such applications and for the formulation of physical theories. The Journal of Mathematical Physics (JMP) features content in all areas of mathematical physics. Specifically, the articles focus on areas of research that illustrate the application of mathematics to problems in physics, the development of mathematical methods for such applications, and for the formulation of physical theories. The mathematics featured in the articles are written so that theoretical physicists can understand them. JMP also publishes review articles on mathematical subjects relevant to physics as well as special issues that combine manuscripts on a topic of current interest to the mathematical physics community. JMP welcomes original research of the highest quality in all active areas of mathematical physics, including the following: Partial Differential Equations Representation Theory and Algebraic Methods Many Body and Condensed Matter Physics Quantum Mechanics - General and Nonrelativistic Quantum Information and Computation Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory General Relativity and Gravitation Dynamical Systems Classical Mechanics and Classical Fields Fluids Statistical Physics Methods of Mathematical Physics.
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